Image coding method based on secondary transform and apparatus therefor

ABSTRACT

An image decoding method according to the present document comprises the steps of: receiving a quantized transform coefficient for a target block and a transform index for a non-separable secondary transform; dequantizing the quantized transform coefficients to derive transform coefficients; deriving modified transform coefficients on the basis of a matrix operation of a transform kernel matrix in a transform set indicated by the transform index and a transform coefficient list corresponding to the magnitude of dequantized transform coefficients; clipping the modified transform coefficients to a predetermined range of values; and deriving residual samples for the target block on the basis of an inverse primary transform with respect to the modified transform coefficients clipped.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.17/349,215, filed on Jun. 16, 2021, which is a continuation pursuant to35 U.S.C. § 119(e) of International Application PCT/KR2020/001533, withan international filing date of Jan. 31, 2020, which claims the benefitof U.S. Provisional Patent Application No. 62/800,384, filed on Feb. 1,2019, the contents of which are hereby incorporated by reference hereinin their entirety.

TECHNICAL FIELD

The present disclosure relates generally to an image coding technologyand, more particularly, to an image coding method based on a transformin an image coding system and an apparatus therefor.

RELATED ART

Nowadays, the demand for high-resolution and high-quality images/videossuch as 4K, 8K or more ultra high definition (UHD) images/videos hasbeen increasing in various fields. As the image/video data becomeshigher resolution and higher quality, the transmitted information amountor bit amount increases as compared to the conventional image data.Therefore, when image data is transmitted using a medium such as aconventional wired/wireless broadband line or image/video data is storedusing an existing storage medium, the transmission cost and the storagecost thereof are increased.

Further, nowadays, the interest and demand for immersive media such asvirtual reality (VR), artificial reality (AR) content or hologram, orthe like is increasing, and broadcasting for images/videos having imagefeatures different from those of real images, such as a game image isincreasing.

Accordingly, there is a need for a highly efficient image/videocompression technique for effectively compressing and transmitting orstoring, and reproducing information of high resolution and high qualityimages/videos having various features as described above.

SUMMARY

A technical aspect of the present disclosure is to provide a method andan apparatus for increasing image coding efficiency.

Another technical aspect of the present disclosure is to provide amethod and an apparatus for increasing transform efficiency.

Still another technical aspect of the present disclosure is to providean image coding method and an image coding apparatus which are based ona clipping of a transform process.

Yet another technical aspect of the present disclosure is to provide amethod and an apparatus for increasing the efficiency of a secondarytransform by changing the array of transform coefficients according toan intra prediction mode.

Still another technical aspect of the present disclosure is to providean image coding method and an image coding apparatus for increasing theefficiency of a secondary transform by optimizing the transformationkernel matrix applied to the secondary transform.

Still another technical aspect of the present disclosure is to providean image coding method and an image coding apparatus which are based ona transform set for increasing coding efficiency.

According to an embodiment of the present disclosure, there is providedan image decoding method performed by a decoding apparatus. The methodmay include: receiving quantized transform coefficients for a targetblock and a transform index for a non-separable secondary transform;deriving transform coefficients by dequantizing the quantized transformcoefficients; deriving the modified transform coefficients based on amatrix operation of a transform kernel matrix in a transform set relatedto the transform index and a transform coefficient list corresponding toa size of dequantized transform coefficients; clipping the modifiedtransform coefficients to values within a predetermined range, andderiving residual samples for the target block based on an inverseprimary transform for clipped modified transform coefficients.

The method may further clip the residual samples to values within apredetermined range .

According to another embodiment of the present disclosure, there isprovided an image encoding method performed by an encoding apparatus.The method may include: deriving prediction samples based on an intraprediction mode applied to a target block; deriving residual samples forthe target block based on the prediction samples; deriving transformcoefficients by applying a primary transform to the residual samples;deriving an input transform coefficient size related to a length of thetransform coefficients to which a non-separable secondary transform isapplied, an output transform coefficient size related to a length ofmodified transform coefficients to which the non-separable secondarytransform has been applied, and a transform set mapped to an intra modefor the target block based on the non-separable secondary transformbeing applied to the transform coefficients; deriving the modifiedtransform coefficients based on a matrix operation of any one transformkernel matrix in the transform set and a transform coefficientcorresponding to the input transform coefficient size; clipping themodified transform coefficients to values within a predetermined range,and deriving quantized transform coefficients by performing quantizationbased on the clipped modified transform coefficients.

According to still another embodiment of the present disclosure, theremay be provided a digital storage medium that stores image dataincluding encoded image information and a bitstream generated accordingto an image encoding method performed by an encoding apparatus.

According to yet another embodiment of the present disclosure, there maybe provided a digital storage medium that stores image data includingencoded image information and a bitstream to cause a decoding apparatusto perform the image decoding method.

According to the present disclosure, it is possible to increase overallimage/video compression efficiency.

According to the present disclosure, it is possible to increase theefficiency of a secondary transform by changing the array of transformcoefficients according to an intra prediction mode.

According to the present disclosure, it is possible to increase imagecoding efficiency by performing image coding based on a clipping of atransform process.

According to the present disclosure, it is possible to increase theefficiency of a secondary transform by optimizing the transformationkernel matrix applied to the secondary transform.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 schematically illustrates an example of a video/image codingsystem to which the present disclosure is applicable.

FIG. 2 is a diagram schematically illustrating a configuration of avideo/image encoding apparatus to which the present disclosure isapplicable.

FIG. 3 is a diagram schematically illustrating a configuration of avideo/image decoding apparatus to which the present disclosure isapplicable.

FIG. 4 schematically illustrates a multiple transform techniqueaccording to an embodiment of the present disclosure.

FIG. 5 illustrates directional intra modes of 65 prediction directions.

FIG. 6 is a diagram illustrating an RST according to an embodiment ofthe present disclosure.

FIG. 7 is a diagram illustrating a transform coefficient scanning orderaccording to an embodiment of the present disclosure.

FIG. 8 is a flowchart illustrating an inverse RST process according toan embodiment of the present disclosure.

FIG. 9 illustrates a forward RST 8×8 using a 16×48 transform matrixaccording to an embodiment of the present disclosure.

FIG. 10 is a flowchart illustrating an operation of a video decodingapparatus according to an embodiment of the present disclosure.

FIG. 11 is a control flowchart illustrating an image decoding method bya decoding apparatus according to an embodiment of the presentdisclosure.

FIG. 12 is a flowchart illustrating an operation of a video encodingapparatus according to an embodiment of the present disclosure.

FIG. 13 is a control flowchart illustrating an image encoding method byan encoding apparatus according to an embodiment of the presentdisclosure.

FIG. 14 illustrates the structure of a content streaming system to whichthe present disclosure is applied.

DESCRIPTION OF EXEMPLARY EMBODIMENTS

While the present disclosure may be susceptible to various modificationsand include various embodiments, specific embodiments thereof have beenshown in the drawings by way of example and will now be described indetail. However, this is not intended to limit the present disclosure tothe specific embodiments disclosed herein. The terminology used hereinis for the purpose of describing specific embodiments only, and is notintended to limit technical idea of the present disclosure. The singularforms may include the plural forms unless the context clearly indicatesotherwise. The terms such as “include” and “have” are intended toindicate that features, numbers, steps, operations, elements,components, or combinations thereof used in the following descriptionexist, and thus should not be understood as that the possibility ofexistence or addition of one or more different features, numbers, steps,operations, elements, components, or combinations thereof is excluded inadvance.

Meanwhile, each component on the drawings described herein isillustrated independently for convenience of description as tocharacteristic functions different from each other, and however, it isnot meant that each component is realized by a separate hardware orsoftware. For example, any two or more of these components may becombined to form a single component, and any single component may bedivided into plural components. The embodiments in which components arecombined and/or divided will belong to the scope of the patent right ofthe present disclosure as long as they do not depart from the essence ofthe present disclosure.

Hereinafter, preferred embodiments of the present disclosure will beexplained in more detail while referring to the attached drawings. Inaddition, the same reference signs are used for the same components onthe drawings, and repeated descriptions for the same components will beomitted.

This document relates to video/image coding. For example, themethod/example disclosed in this document may relate to a VVC (VersatileVideo Coding) standard (ITU-T Rec. H.266), a next-generation video/imagecoding standard after VVC, or other video coding related standards (e.g., HEVC (High Efficiency Video Coding) standard (ITU-T Rec. H.265), EVC(essential video coding) standard, AVS2 standard, etc.).

In this document, a variety of embodiments relating to video/imagecoding may be provided, and, unless specified to the contrary, theembodiments may be combined to each other and be performed.

In this document, a video may mean a set of a series of images overtime. Generally a picture means a unit representing an image at aspecific time zone, and a slice/tile is a unit constituting a part ofthe picture. The slice/tile may include one or more coding tree units(CTUs). One picture may be constituted by one or more slices/tiles. Onepicture may be constituted by one or more tile groups. One tile groupmay include one or more tiles.

A pixel or a pel may mean a smallest unit constituting one picture (orimage). Also, ‘sample’ may be used as a term corresponding to a pixel. Asample may generally represent a pixel or a value of a pixel, and mayrepresent only a pixel/pixel value of a luma component or only apixel/pixel value of a chroma component. Alternatively, the sample mayrefer to a pixel value in the spatial domain, or when this pixel valueis converted to the frequency domain, it may refer to a transformcoefficient in the frequency domain.

A unit may represent the basic unit of image processing. The unit mayinclude at least one of a specific region and information related to theregion. One unit may include one luma block and two chroma (e.g., cb andcr) blocks. The unit and a term such as a block, an region, or the likemay be used in place of each other according to circumstances. In ageneral case, an M×N block may include a set (or an array) of samples(or sample arrays) or transform coefficients consisting of M columns andN rows.

In this document, the term “/” and “,” should be interpreted to indicate“and/or.” For instance, the expression “A/B” may mean “A and/or B.”Further, “A, B” may mean “A and/or B.” Further, “A/B/C” may mean “atleast one of A, B, and/or C.” Also, “A/B/C” may mean “at least one of A,B, and/or C.”

Further, in the document, the term “or” should be interpreted toindicate “and/or.” For instance, the expression “A or B” may include 1)only A, 2) only B, and/or 3) both A and B. In other words, the term “or”in this document should be interpreted to indicate “additionally oralternatively.”

FIG. 1 schematically illustrates an example of a video/image codingsystem to which the present disclosure is applicable.

Referring to FIG. 1, the video/image coding system may include a firstdevice (source device) and a second device (receive device). The sourcedevice may deliver encoded video/image information or data in the formof a file or streaming to the receive device via a digital storagemedium or network.

The source device may include a video source, an encoding apparatus, anda transmitter. The receive device may include a receiver, a decodingapparatus, and a renderer. The encoding apparatus may be called avideo/image encoding apparatus, and the decoding apparatus may be calleda video/image decoding apparatus. The transmitter may be included in theencoding apparatus. The receiver may be included in the decodingapparatus. The renderer may include a display, and the display may beconfigured as a separate device or an external component.

The video source may obtain a video/image through a process ofcapturing, synthesizing, or generating a video/image. The video sourcemay include a video/image capture device and/or a video/image generatingdevice. The video/image capture device may include, for example, one ormore cameras, video/image archives including previously capturedvideo/images, or the like. The video/image generating device mayinclude, for example, a computer, a tablet and a smartphone, and may(electronically) generate a video/image. For example, a virtualvideo/image may be generated through a computer or the like. In thiscase, the video/image capturing process may be replaced by a process ofgenerating related data.

The encoding apparatus may encode an input video/image. The encodingapparatus may perform a series of procedures such as prediction,transform, and quantization for compression and coding efficiency. Theencoded data (encoded video/image information) may be output in the formof a bitstream.

The transmitter may transmit the encoded video/image information or dataoutput in the form of a bitstream to the receiver of the receive devicethrough a digital storage medium or a network in the form of a file orstreaming. The digital storage medium may include various storagemediums such as USB, SD, CD, DVD, Blu-ray, HDD, SSD, and the like. Thetransmitter may include an element for generating a media file through apredetermined file format, and may include an element for transmissionthrough a broadcast/communication network. The receiver mayreceive/extract the bitstream and transmit the received/extractedbitstream to the decoding apparatus.

The decoding apparatus may decode a video/image by performing a seriesof procedures such as dequantization, inverse transform, prediction, andthe like corresponding to the operation of the encoding apparatus.

The renderer may render the decoded video/image. The renderedvideo/image may be displayed through the display.

FIG. 2 is a diagram schematically illustrating a configuration of avideo/image encoding apparatus to which the present disclosure isapplicable. Hereinafter, what is referred to as the video encodingapparatus may include an image encoding apparatus.

Referring to FIG. 2, the encoding apparatus 200 may include an imagepartitioner 210, a predictor 220, a residual processor 230, an entropyencoder 240, an adder 250, a filter 260, and a memory 270. The predictor220 may include an inter predictor 221 and an intra predictor 222. Theresidual processor 230 may include a transformer 232, a quantizer 233, adequantizer 234, an inverse transformer 235. The residual processor 230may further include a subtractor 231. The adder 250 may be called areconstructor or a reconstructed block generator. The image partitioner210, the predictor 220, the residual processor 230, the entropy encoder240, the adder 250, and the filter 260, which have been described above,may be constituted by one or more hardware components (e.g., encoderchipsets or processors) according to an embodiment. Further, the memory270 may include a decoded picture buffer (DPB), and may be constitutedby a digital storage medium. The hardware component may further includethe memory 270 as an internal/external component.

The image partitioner 210 may partition an input image (or a picture ora frame) input to the encoding apparatus 200 into one or more processingunits. As one example, the processing unit may be called a coding unit(CU). In this case, starting with a coding tree unit (CTU) or thelargest coding unit (LCU), the coding unit may be recursivelypartitioned according to the Quad-tree binary-tree ternary-tree (QTBTTT)structure. For example, one coding unit may be divided into a pluralityof coding units of a deeper depth based on the quad-tree structure, thebinary-tree structure, and/or the ternary structure. In this case, forexample, the quad-tree structure may be applied first and thebinary-tree structure and/or the ternary structure may be applied later.Alternatively, the binary-tree structure may be applied first. Thecoding procedure according to the present disclosure may be performedbased on the final coding unit which is not further partitioned. In thiscase, the maximum coding unit may be used directly as a final codingunit based on coding efficiency according to the image characteristic.Alternatively, the coding unit may be recursively partitioned intocoding units of a further deeper depth as needed, so that the codingunit of an optimal size may be used as a final coding unit. Here, thecoding procedure may include procedures such as prediction, transform,and reconstruction, which will be described later. As another example,the processing unit may further include a prediction unit (PU) or atransform unit (TU). In this case, the prediction unit and the transformunit may be split or partitioned from the above-described final codingunit. The prediction unit may be a unit of sample prediction, and thetransform unit may be a unit for deriving a transform coefficient and/ora unit for deriving a residual signal from a transform coefficient.

The unit and a term such as a block, an region, or the like may be usedin place of each other according to circumstances. In a general case, anM×N block may represent a set of samples or transform coefficientsconsisting of M columns and N rows. The sample may generally represent apixel or a value of a pixel, and may represent only a pixel/pixel valueof a luma component, or only a pixel/pixel value of a chroma component.The sample may be used as a term corresponding to a pixel or a pel ofone picture (or image).

The subtractor 231 subtracts a prediction signal (predicted block,prediction sample array) output from the inter predictor 221 or theintra predictor 222 from an input image signal (original block, originalsample array) to generate a residual signal (residual block, residualsample array), and the generated residual signal is transmitted to thetransformer 232. In this case, as shown, a unit which subtracts theprediction signal (predicted block, prediction sample array) from theinput image signal (original block, original sample array) in theencoder 200 may be called the subtractor 231. The predictor may performprediction on a processing target block (hereinafter, referred to as‘current block’), and may generate a predicted block includingprediction samples for the current block. The predictor may determinewhether intra prediction or inter prediction is applied on a currentblock or CU basis. As discussed later in the description of eachprediction mode, the predictor may generate various pieces ofinformation relating to prediction, such as prediction mode information,and transmit the generated information to the entropy encoder 240. Theinformation on the prediction may be encoded in the entropy encoder 240and output in the form of a bitstream.

The intra predictor 222 may predict the current block by referring tosamples in the current picture. The referred samples may be located inthe neighbor of or apart from the current block according to theprediction mode. In the intra prediction, prediction modes may include aplurality of non-directional modes and a plurality of directional modes.The non-directional modes may include, for example, a DC mode and aplanar mode. The directional mode may include, for example, 33directional prediction modes or 65 directional prediction modesaccording to the degree of detail of the prediction direction. However,this is merely an example, and more or less directional prediction modesmay be used depending on a setting. The intra predictor 222 maydetermine the prediction mode applied to the current block by using theprediction mode applied to the neighboring block.

The inter predictor 221 may derive a predicted block for the currentblock based on a reference block (reference sample array) specified by amotion vector on a reference picture. At this time, in order to reducethe amount of motion information transmitted in the inter predictionmode, the motion information may be predicted on a block, subblock, orsample basis based on correlation of motion information between theneighboring block and the current block. The motion information mayinclude a motion vector and a reference picture index. The motioninformation may further include inter prediction direction (L0prediction, L1 prediction, Bi prediction, etc.) information. In the caseof inter prediction, the neighboring block may include a spatialneighboring block existing in the current picture and a temporalneighboring block existing in the reference picture. The referencepicture including the reference block and the reference pictureincluding the temporal neighboring block may be same to each other ordifferent from each other. The temporal neighboring block may be calleda collocated reference block, a collocated CU (colCU), and the like, andthe reference picture including the temporal neighboring block may becalled a collocated picture (colPic). For example, the inter predictor221 may configure a motion information candidate list based onneighboring blocks and generate information indicating which candidateis used to derive a motion vector and/or a reference picture index ofthe current block. Inter prediction may be performed based on variousprediction modes. For example, in the case of a skip mode and a mergemode, the inter predictor 221 may use motion information of theneighboring block as motion information of the current block. In theskip mode, unlike the merge mode, the residual signal may not betransmitted. In the case of the motion information prediction (motionvector prediction, MVP) mode, the motion vector of the neighboring blockmay be used as a motion vector predictor and the motion vector of thecurrent block may be indicated by signaling a motion vector difference.

The predictor 220 may generate a prediction signal based on variousprediction methods. For example, the predictor may apply intraprediction or inter prediction for prediction on one block, and, aswell, may apply intra prediction and inter prediction at the same time.This may be called combined inter and intra prediction (CIIP). Further,the predictor may be based on an intra block copy (IBC) prediction mode,or a palette mode in order to perform prediction on a block. The IBCprediction mode or palette mode may be used for content image/videocoding of a game or the like, such as screen content coding (SCC).Although the IBC basically performs prediction in a current block, itcan be performed similarly to inter prediction in that it derives areference block in a current block. That is, the IBC may use at leastone of inter prediction techniques described in the present disclosure.

The prediction signal generated through the inter predictor 221 and/orthe intra predictor 222 may be used to generate a reconstructed signalor to generate a residual signal. The transformer 232 may generatetransform coefficients by applying a transform technique to the residualsignal. For example, the transform technique may include at least one ofa discrete cosine transform (DCT), a discrete sine transform (DST), aKarhunen-Loeve transform (KLT), a graph-based transform (GBT), or aconditionally non-linear transform (CNT). Here, the GBT means transformobtained from a graph when relationship information between pixels isrepresented by the graph. The CNT refers to transform obtained based ona prediction signal generated using all previously reconstructed pixels.In addition, the transform process may be applied to square pixel blockshaving the same size or may be applied to blocks having a variable sizerather than the square one.

The quantizer 233 may quantize the transform coefficients and transmitthem to the entropy encoder 240, and the entropy encoder 240 may encodethe quantized signal (information on the quantized transformcoefficients) and output the encoded signal in a bitstream. Theinformation on the quantized transform coefficients may be referred toas residual information. The quantizer 233 may rearrange block typequantized transform coefficients into a one-dimensional vector formbased on a coefficient scan order, and generate information on thequantized transform coefficients based on the quantized transformcoefficients of the one-dimensional vector form. The entropy encoder 240may perform various encoding methods such as, for example, exponentialGolomb, context-adaptive variable length coding (CAVLC),context-adaptive binary arithmetic coding (CABAC), and the like. Theentropy encoder 240 may encode information necessary for video/imagereconstruction other than quantized transform coefficients (e.g. valuesof syntax elements, etc.) together or separately. Encoded information(e.g., encoded video/image information) may be transmitted or stored ona unit basis of a network abstraction layer (NAL) in the form of abitstream. The video/image information may further include informationon various parameter sets such as an adaptation parameter set (APS), apicture parameter set (PPS), a sequence parameter set (SPS), a videoparameter set (VPS) or the like. Further, the video/image informationmay further include general constraint information. In the presentdisclosure, information and/or syntax elements which aretransmitted/signaled to the decoding apparatus from the encodingapparatus may be included in video/image information. The video/imageinformation may be encoded through the above-described encodingprocedure and included in the bitstream. The bitstream may betransmitted through a network, or stored in a digital storage medium.Here, the network may include a broadcast network, a communicationnetwork and/or the like, and the digital storage medium may includevarious storage media such as USB, SD, CD, DVD, Blu-ray, HDD, SSD, andthe like. A transmitter (not shown) which transmits a signal output fromthe entropy encoder 240 and/or a storage (not shown) which stores it maybe configured as an internal/external element of the encoding apparatus200, or the transmitter may be included in the entropy encoder 240.

Quantized transform coefficients output from the quantizer 233 may beused to generate a prediction signal. For example, by applyingdequantization and inverse transform to quantized transform coefficientsthrough the dequantizer 234 and the inverse transformer 235, theresidual signal (residual block or residual samples) may bereconstructed. The adder 155 adds the reconstructed residual signal to aprediction signal output from the inter predictor 221 or the intrapredictor 222, so that a reconstructed signal (reconstructed picture,reconstructed block, reconstructed sample array) may be generated. Whenthere is no residual for a processing target block as in a case wherethe skip mode is applied, the predicted block may be used as areconstructed block. The adder 250 may be called a reconstructor or areconstructed block generator. The generated reconstructed signal may beused for intra prediction of a next processing target block in thecurrent block, and as described later, may be used for inter predictionof a next picture through filtering.

Meanwhile, in the picture encoding and/or reconstructing process, lumamapping with chroma scaling (LMCS) may be applied.

The filter 260 may improve subjective/objective video quality byapplying the filtering to the reconstructed signal. For example, thefilter 260 may generate a modified reconstructed picture by applyingvarious filtering methods to the reconstructed picture, and may storethe modified reconstructed picture in the memory 270, specifically inthe DPB of the memory 270. The various filtering methods may include,for example, deblocking filtering, sample adaptive offset, an adaptiveloop filter, a bilateral filter or the like. As discussed later in thedescription of each filtering method, the filter 260 may generatevarious pieces of information relating to filtering, and transmit thegenerated information to the entropy encoder 240. The information on thefiltering may be encoded in the entropy encoder 240 and output in theform of a bitstream.

The modified reconstructed picture which has been transmitted to thememory 270 may be used as a reference picture in the inter predictor221. Through this, the encoding apparatus can avoid prediction mismatchin the encoding apparatus 100 and a decoding apparatus when the interprediction is applied, and can also improve coding efficiency.

The DPB of the memory 270 may store the modified reconstructed picturein order to use it as a reference picture in the inter predictor 221.The memory 270 may store motion information of a block in the currentpicture, from which motion information has been derived (or encoded)and/or motion information of blocks in an already reconstructed picture.The stored motion information may be transmitted to the inter predictor221 to be utilized as motion information of a neighboring block ormotion information of a temporal neighboring block. The memory 270 maystore reconstructed samples of reconstructed blocks in the currentpicture, and transmit them to the intra predictor 222.

FIG. 3 is a diagram schematically illustrating a configuration of avideo/image decoding apparatus to which the present disclosure isapplicable.

Referring to FIG. 3, the video decoding apparatus 300 may include anentropy decoder 310, a residual processor 320, a predictor 330, an adder340, a filter 350 and a memory 360. The predictor 330 may include aninter predictor 331 and an intra predictor 332. The residual processor320 may include a dequantizer 321 and an inverse transformer 321. Theentropy decoder 310, the residual processor 320, the predictor 330, theadder 340, and the filter 350, which have been described above, may beconstituted by one or more hardware components (e.g., decoder chipsetsor processors) according to an embodiment. Further, the memory 360 mayinclude a decoded picture buffer (DPB), and may be constituted by adigital storage medium. The hardware component may further include thememory 360 as an internal/external component.

When a bitstream including video/image information is input, thedecoding apparatus 300 may reconstruct an image correspondingly to aprocess by which video/image information has been processed in theencoding apparatus of FIG. 2. For example, the decoding apparatus 300may derive units/blocks based on information relating to block partitionobtained from the bitstream. The decoding apparatus 300 may performdecoding by using a processing unit applied in the encoding apparatus.Therefore, the processing unit of decoding may be, for example, a codingunit, which may be partitioned along the quad-tree structure, thebinary-tree structure, and/or the ternary-tree structure from a codingtree unit or a largest coding unit. One or more transform units may bederived from the coding unit. And, the reconstructed image signaldecoded and output through the decoding apparatus 300 may be reproducedthrough a reproducer.

The decoding apparatus 300 may receive a signal output from the encodingapparatus of FIG. 2 in the form of a bitstream, and the received signalmay be decoded through the entropy decoder 310. For example, the entropydecoder 310 may parse the bitstream to derive information (e.g.,video/image information) required for image reconstruction (or picturereconstruction). The video/image information may further includeinformation on various parameter sets such as an adaptation parameterset (APS), a picture parameter set (PPS), a sequence parameter set(SPS), a video parameter set (VPS) or the like. Further, the video/imageinformation may further include general constraint information. Thedecoding apparatus may decode a picture further based on information onthe parameter set and/or the general constraint information. In thepresent disclosure, signaled/received information and/or syntaxelements, which will be described later, may be decoded through thedecoding procedure and be obtained from the bitstream. For example, theentropy decoder 310 may decode information in the bitstream based on acoding method such as exponential Golomb encoding, CAVLC, CABAC, or thelike, and may output a value of a syntax element necessary for imagereconstruction and quantized values of a transform coefficient regardinga residual. More specifically, a CABAC entropy decoding method mayreceive a bin corresponding to each syntax element in a bitstream,determine a context model using decoding target syntax elementinformation and decoding information of neighboring and decoding targetblocks, or information of symbol/bin decoded in a previous step, predictbin generation probability according to the determined context model andperform arithmetic decoding of the bin to generate a symbolcorresponding to each syntax element value. Here, the CABAC entropydecoding method may update the context model using information of asymbol/bin decoded for a context model of the next symbol/bin afterdetermination of the context model. Information on prediction amonginformation decoded in the entropy decoder 310 may be provided to thepredictor (inter predictor 332 and intra predictor 331), and residualvalues, that is, quantized transform coefficients, on which entropydecoding has been performed in the entropy decoder 310, and associatedparameter information may be input to the residual processor 320. Theresidual processor 320 may derive a residual signal (residual block,residual sample, or residual sample array). Further, information onfiltering among information decoded in the entropy decoder 310 may beprovided to the filter 350. Meanwhile, a receiver (not shown) whichreceives a signal output from the encoding apparatus may furtherconstitute the decoding apparatus 300 as an internal/external element,and the receiver may be a component of the entropy decoder 310.Meanwhile, the decoding apparatus according to the present disclosuremay be called a video/image/picture coding apparatus, and the decodingapparatus may be classified into an information decoder(video/image/picture information decoder) and a sample decoder(video/image/picture sample decoder). The information decoder mayinclude the entropy decoder 310, and the sample decoder may include atleast one of the dequantizer 321, the inverse transformer 322, the adder340, the filter 350, the memory 360, the inter predictor 332, and theintra predictor 331.

The dequantizer 321 may output transform coefficients by dequantizingthe quantized transform coefficients. The dequantizer 321 may rearrangethe quantized transform coefficients in the form of a two-dimensionalblock. In this case, the rearrangement may perform rearrangement basedon an order of coefficient scanning which has been performed in theencoding apparatus. The dequantizer 321 may perform dequantization onthe quantized transform coefficients using quantization parameter (e.g.,quantization step size information), and obtain transform coefficients.

The deqauntizer 322 obtains a residual signal (residual block, residualsample array) by inverse transforming transform coefficients.

The predictor may perform prediction on the current block, and generatea predicted block including prediction samples for the current block.The predictor may determine whether intra prediction or inter predictionis applied to the current block based on the information on predictionoutput from the entropy decoder 310, and specifically may determine anintra/inter prediction mode.

The predictor may generate a prediction signal based on variousprediction methods. For example, the predictor may apply intraprediction or inter prediction for prediction on one block, and, aswell, may apply intra prediction and inter prediction at the same time.This may be called combined inter and intra prediction (CIIP). Inaddition, the predictor may perform intra block copy (IBC) forprediction on a block. The intra block copy may be used for contentimage/video coding of a game or the like, such as screen content coding(SCC). Although the IBC basically performs prediction in a currentblock, it can be performed similarly to inter prediction in that itderives a reference block in a current block. That is, the IBC may useat least one of inter prediction techniques described in the presentdisclosure.

The intra predictor 331 may predict the current block by referring tothe samples in the current picture. The referred samples may be locatedin the neighbor of or apart from the current block according to theprediction mode. In the intra prediction, prediction modes may include aplurality of non-directional modes and a plurality of directional modes.The intra predictor 331 may determine the prediction mode applied to thecurrent block by using the prediction mode applied to the neighboringblock.

The inter predictor 332 may derive a predicted block for the currentblock based on a reference block (reference sample array) specified by amotion vector on a reference picture. At this time, in order to reducethe amount of motion information transmitted in the inter predictionmode, the motion information may be predicted on a block, subblock, orsample basis based on correlation of motion information between theneighboring block and the current block. The motion information mayinclude a motion vector and a reference picture index. The motioninformation may further include inter prediction direction (L0prediction, L1 prediction, Bi prediction, etc.) information. In the caseof inter prediction, the neighboring block may include a spatialneighboring block existing in the current picture and a temporalneighboring block existing in the reference picture. For example, theinter predictor 332 may configure a motion information candidate listbased on neighboring blocks, and derive a motion vector and/or areference picture index of the current block based on received candidateselection information. Inter prediction may be performed based onvarious prediction modes, and the information on prediction may includeinformation indicating a mode of inter prediction for the current block.

The adder 340 may generate a reconstructed signal (reconstructedpicture, reconstructed block, reconstructed sample array) by adding theobtained residual signal to the prediction signal (predicted block,prediction sample array) output from the predictor 330. When there is noresidual for a processing target block as in a case where the skip modeis applied, the predicted block may be used as a reconstructed block.

The adder 340 may be called a reconstructor or a reconstructed blockgenerator. The generated reconstructed signal may be used for intraprediction of a next processing target block in the current block, andas described later, may be output through filtering or be used for interprediction of a next picture.

Meanwhile, in the picture decoding process, luma mapping with chromascaling (LMCS) may be applied.

The filter 350 may improve subjective/objective video quality byapplying the filtering to the reconstructed signal. For example, thefilter 350 may generate a modified reconstructed picture by applyingvarious filtering methods to the reconstructed picture, and may transmitthe modified reconstructed picture in the memory 360, specifically inthe DPB of the memory 360. The various filtering methods may include,for example, deblocking filtering, sample adaptive offset, an adaptiveloop filter, a bilateral filter or the like.

The (modified) reconstructed picture which has been stored in the DPB ofthe memory 360 may be used as a reference picture in the inter predictor332. The memory 360 may store motion information of a block in thecurrent picture, from which motion information has been derived (ordecoded) and/or motion information of blocks in an already reconstructedpicture. The stored motion information may be transmitted to the interpredictor 260 to be utilized as motion information of a neighboringblock or motion information of a temporal neighboring block. The memory360 may store reconstructed samples of reconstructed blocks in thecurrent picture, and transmit them to the intra predictor 331.

In this specification, the examples described in the predictor 330, thedequantizer 321, the inverse transformer 322, and the filter 350 of thedecoding apparatus 300 may be similarly or correspondingly applied tothe predictor 220, the dequantizer 234, the inverse transformer 235, andthe filter 260 of the encoding apparatus 200, respectively.

As described above, prediction is performed in order to increasecompression efficiency in performing video coding. Through this, apredicted block including prediction samples for a current block, whichis a coding target block, may be generated. Here, the predicted blockincludes prediction samples in a space domain (or pixel domain). Thepredicted block may be identically derived in the encoding apparatus andthe decoding apparatus, and the encoding apparatus may increase imagecoding efficiency by signaling to the decoding apparatus not originalsample value of an original block itself but information on residual(residual information) between the original block and the predictedblock. The decoding apparatus may derive a residual block includingresidual samples based on the residual information, generate areconstructed block including reconstructed samples by adding theresidual block to the predicted block, and generate a reconstructedpicture including reconstructed blocks.

The residual information may be generated through transform andquantization procedures. For example, the encoding apparatus may derivea residual block between the original block and the predicted block,derive transform coefficients by performing a transform procedure onresidual samples (residual sample array) included in the residual block,and derive quantized transform coefficients by performing a quantizationprocedure on the transform coefficients, so that it may signalassociated residual information to the decoding apparatus (through abitstream). Here, the residual information may include valueinformation, position information, a transform technique, transformkernel, a quantization parameter or the like of the quantized transformcoefficients. The decoding apparatus may perform aquantization/dequantization procedure and derive the residual samples(or residual sample block), based on residual information. The decodingapparatus may generate a reconstructed block based on a predicted blockand the residual block. The encoding apparatus may derive a residualblock by dequantizing/inverse transforming quantized transformcoefficients for reference for inter prediction of a next picture, andmay generate a reconstructed picture based on this.

FIG. 4 schematically illustrates a multiple transform techniqueaccording to an embodiment of the present disclosure.

Referring to FIG. 4, a transformer may correspond to the transformer inthe encoding apparatus of foregoing FIG. 2, and an inverse transformermay correspond to the inverse transformer in the encoding apparatus offoregoing FIG. 2, or to the inverse transformer in the decodingapparatus of FIG. 3.

The transformer may derive (primary) transform coefficients byperforming a primary transform based on residual samples (residualsample array) in a residual block (S410). This primary transform may bereferred to as a core transform. Herein, the primary transform may bebased on multiple transform selection (MTS), and when a multipletransform is applied as the primary transform, it may be referred to asa multiple core transform.

The multiple core transform may represent a method of transformingadditionally using discrete cosine transform (DCT) type 2 and discretesine transform (DST) type 7, DCT type 8, and/or DST type 1. That is, themultiple core transform may represent a transform method of transforminga residual signal (or residual block) of a space domain into transformcoefficients (or primary transform coefficients) of a frequency domainbased on a plurality of transform kernels selected from among the DCTtype 2, the DST type 7, the DCT type 8 and the DST type 1. Herein, theprimary transform coefficients may be called temporary transformcoefficients from the viewpoint of the transformer.

In other words, when the conventional transform method is applied,transform coefficients might be generated by applying transform from aspace domain to a frequency domain for a residual signal (or residualblock) based on the DCT type 2. Unlike to this, when the multiple coretransform is applied, transform coefficients (or primary transformcoefficients) may be generated by applying transform from a space domainto a frequency domain for a residual signal (or residual block) based onthe DCT type 2, the DST type 7, the DCT type 8, and/or DST type 1.Herein, the DCT type 2, the DST type 7, the DCT type 8, and the DST type1 may be called a transform type, transform kernel or transform core.

For reference, the DCT/DST transform types may be defined based on basisfunctions, and the basis functions may be represented as in thefollowing table.

TABLE 1 Transform Type Basis function T_(i)(j) i, j = 0, 1, . . . , N-1DCT-II${T_{i}(j)} = {\omega_{0} \cdot \sqrt{\frac{2}{N}} \cdot {\cos\left( \frac{\pi \cdot i \cdot \left( {{2j} + 1} \right)}{2N} \right)}}$${{where}\mspace{14mu}\omega_{0}} = \left\{ \begin{matrix}\sqrt{\frac{2}{N}} & {i = 0} \\1 & {i \neq 0}\end{matrix} \right.$ DCT-V${{T_{i}(j)} = {\omega_{0} \cdot \omega_{1} \cdot \sqrt{\frac{2}{{2N} - 1}} \cdot {\cos\left( \frac{2{\pi \cdot i \cdot j}}{{2N} - 1} \right)}}},$${{where}\mspace{14mu}\omega_{0}} = \left\{ {{\begin{matrix}\sqrt{\frac{2}{N}} & {i = 0} \\1 & {i \neq 0}\end{matrix}.\omega_{1}} = \left\{ \begin{matrix}\sqrt{\frac{2}{N}} & {j = 0} \\1 & {j \neq 0}\end{matrix} \right.} \right.$ DCT-VIII${T_{i}(j)} = {{\sqrt{\frac{4}{{2N} + 1}} \cdot \cos}\;\left( \frac{\pi \cdot \left( {{2i} + 1} \right) \cdot \left( {{2j} + 1} \right)}{{4N} + 2} \right)}$DST-I${T_{i}(j)} = {{\sqrt{\frac{2}{N + 1}} \cdot \sin}\;\left( \frac{\pi \cdot \left( {i + 1} \right) \cdot \left( {j + 1} \right)}{N + 1} \right)}$DST-VII${T_{i}(j)} = {{\sqrt{\frac{4}{{2N} + 1}} \cdot \sin}\;\left( \frac{\pi \cdot \left( {{2i} + 1} \right) \cdot \left( {j + 1} \right)}{{2N} + 1} \right)}$

If the multiple core transform is performed, then a vertical transformkernel and a horizontal transform kernel for a target block may beselected from among the transform kernels, a vertical transform for thetarget block may be performed based on the vertical transform kernel,and a horizontal transform for the target block may be performed basedon the horizontal transform kernel. Here, the horizontal transform mayrepresent a transform for horizontal components of the target block, andthe vertical transform may represent a transform for vertical componentsof the target block. The vertical transform kernel/horizontal transformkernel may be adaptively determined based on a prediction mode and/or atransform index of a target block (CU or sub-block) including a residualblock.

Further, according to an example, if the primary transform is performedby applying the MTS, a mapping relationship for transform kernels may beset by setting specific basis functions to predetermined values andcombining basis functions to be applied in the vertical transform or thehorizontal transform. For example, when the horizontal transform kernelis expressed as trTypeHor and the vertical direction transform kernel isexpressed as trTypeVer, a trTypeHor or trTypeVer value of 0 may be setto DCT2, a trTypeHor or trTypeVer value of 1 may be set to DST7, and atrTypeHor or trTypeVer value of 2 may be set to DCT8.

In this case, MTS index information may be encoded and signaled to thedecoding apparatus to indicate any one of a plurality of transformkernel sets. For example, an MTS index of 0 may indicate that bothtrTypeHor and trTypeVer values are 0, an MTS index of 1 may indicatethat both trTypeHor and trTypeVer values are 1, an MTS index of 2 mayindicatethat the trTypeHor value is 2 and the trTypeVer value. Is 1, anMTS index of 3 may indicate that the trTypeHor value is 1 and thetrTypeVer value is 2, and an MTS index of 4 may indicate that both bothtrTypeHor and trTypeVer values are 2.

The transformer may derive modified (secondary) transform coefficientsby performing the secondary transform based on the (primary) transformcoefficients (S420). The primary transform is a transform from a spatialdomain to a frequency domain, and the secondary transform refers totransforming into a more compressive expression by using a correlationexisting between (primary) transform coefficients. The secondarytransform may include a non-separable transform. In this case, thesecondary transform may be called a non-separable secondary transform(NSST), or a mode-dependent non-separable secondary transform (MDNSST).The non-separable secondary transform may represent a transform whichgenerates modified transform coefficients (or secondary transformcoefficients) for a residual signal by secondary-transforming, based ona non-separable transform matrix, (primary) transform coefficientsderived through the primary transform. At this time, the verticaltransform and the horizontal transform may not be applied separately (orhorizontal and vertical transforms may not be applied independently) tothe (primary) transform coefficients, but the transforms may be appliedat once based on the non-separable transform matrix. In other words, thenon-separable secondary transform may represent a transform method inwhich the vertical and horizontal components of the (primary) transformcoefficients are not separated, and for example, two-dimensional signals(transform coefficients) are re-arranged to a one-dimensional signalthrough a certain determined direction (e.g., row-first direction orcolumn-first direction), and then modified transform coefficients (orsecondary transform coefficients) are generated based on thenon-separable transform matrix. For example, according to a row-firstorder, M×N blocks are disposed in a line in an order of a first row, asecond row, . . . , and an Nth row. According to a column-first order,M×N blocks are disposed in a line in an order of a first column, asecond column, . . . , and an Nth column. The non-separable secondarytransform may be applied to a top-left region of a block configured with(primary) transform coefficients (hereinafter, may be referred to as atransform coefficient block). For example, if the width (W) and theheight (H) of the transform coefficient block are all equal to orgreater than 8, an 8×8 non-separable secondary transform may be appliedto a top-left 8×8 region of the transform coefficient block. Further, ifthe width (W) and the height (H) of the transform coefficient block areall equal to or greater than 4, and the width (W) or the height (H) ofthe transform coefficient block is less than 8, then a 4×4 non-separablesecondary transform may be applied to a top-left min(8,W)×min(8,H)region of the transform coefficient block. However, the embodiment isnot limited to this, and for example, even if only the condition thatthe width (W) or height (H) of the transform coefficient block is equalto or greater than 4 is satisfied, the 4×4 non-separable secondarytransform may be applied to the top-left min(8,W)×min(8,H) region of thetransform coefficient block.

Specifically, for example, if a 4×4 input block is used, thenon-separable secondary transform may be performed as follows.

The 4×4 input block X may be represented as follows.

$\begin{matrix}{X = \begin{bmatrix}X_{00} & X_{01} & X_{02} & X_{03} \\X_{10} & X_{11} & X_{12} & X_{13} \\X_{20} & X_{21} & X_{22} & X_{23} \\X_{30} & X_{31} & X_{32} & X_{33}\end{bmatrix}} & \left\lbrack {{Equation}\mspace{14mu} 1} \right\rbrack\end{matrix}$

If the X is represented in the form of a vector, the vector

may be represented as below.

=[X ₀₀ X ₀₁ X ₀₂ X ₀₃ X ₁₀ X ₁₁ X ₁₂ X ₁₃ X ₂₀ X ₂₁ X ₂₂ X ₂₃ X ₃₀ X ₃₁X ₃₂ X ₃₃]^(T)   [Equation 2]

In Equation 2, the vector X is a one-dimensional vector obtained byrearranging the two-dimensional block X of Equation 1 according to therow-first order.

In this case, the secondary non-separable transform may be calculated asbelow.

=T·

  [Equation 3]

In this equation,

represents a transform coefficient vector, and T represents a 16 ×16(non-separable) transform matrix.

Through foregoing Equation 3, a 16×1 transform coefficient vector

may be derived, and the

may be re-organized into a 4×4 block through a scan order (horizontal,vertical, diagonal and the like). However, the above-describedcalculation is an example, and hypercube-Givens transform (HyGT) or thelike may be used for the calculation of the non-separable secondarytransform in order to reduce the computational complexity of thenon-separable secondary transform.

Meanwhile, in the non-separable secondary transform, a transform kernel(or transform core, transform type) may be selected to bemode-dependent. In this case, the mode may include the intra predictionmode and/or the inter prediction mode.

As described above, the non-separable secondary transform may beperformed based on an 8×8 transform or a 4×4 transform determined basedon the width (W) and the height (H) of the transform coefficient block.The 8×8 transform refers to a transform that is applicable to an 8×8region included in the transform coefficient block when both W and H areequal to or greater than 8, and the 8×8 region may be a top-left 8×8region in the transform coefficient block. Similarly, the 4×4 transformrefers to a transform that is applicable to a 4×4 region included in thetransform coefficient block when both W and H are equal to or greaterthan 4, and the 4×4 region may be a top-left 4×4 region in the transformcoefficient block. . For example, an 8×8 transform kernel matrix may bea 64×64/16×64 matrix, and a 4×4 transform kernel matrix may be a16×16/8×16 matrix.

Here, to select a mode-based transform kernel, three non-separablesecondary transform kernels may be configured per transform set for thenon-separable secondary transform for both the 8×8 transform and the 4×4transform, and there may be 35 transform sets. That is, 35 transformsets may be configured for the 8×8 transform, and 35 transform sets maybe configured for the 4×4 transform. In this case, three 8×8 transformkernels may be included in each of the 35 transform sets for the 8×8transform, and three4×4 transform kernels may be included in each of the35 transform sets for the 4×4 transform. The sizes of the transforms,the numbers of sets, and the numbers of transform kernels in each setmentioned above are merely for illustration. Instead, a size other than8×8 or 4×4 may be used, n sets may be configured, and k transformkernels may be included in each set.

The transform set may be called an NSST set, and the transform kernel inthe NSST set may be called an NSST kernel. The selection of a specificset from among the transform sets may be performed, for example, basedon the intra prediction mode of the target block (CU or sub-block).

For reference, as an example, the intra prediction mode may include twonon-directional (or non-angular) intra prediction modes and 65directional (or angular) intra prediction modes. The non-directionalintra prediction modes may include a planar intra prediction mode, whichis intra prediction mode 0, and a DC intra prediction mode, which isintra prediction mode 1, and the directional intra prediction modes mayinclude 65 intra prediction modes from intra prediction mode 2 to intraprediction mode 66. However, this is an example, and the presentdisclosure may be applied to cases where there are different numbers ofintra prediction modes. According to circumstances, intra predictionmode 67 may be further used, and intra prediction mode 67 may representa linear model (LM) mode.

FIG. 5 illustrates directional intra modes in 65 prediction directions.

Referring to FIG. 5, on the basis of intra prediction mode 34 having aleft upward diagonal prediction direction, the intra prediction modehaving a horizontal directionality and the intra prediction mode havingvertical directionality may be classified. H and V of FIG. 5 meanhorizontal directionality and vertical directionality, respectively, andnumerals −32 to 32 indicate displacements in 1/32 units on the samplegrid position. This may represent an offset for the mode index value.Intra prediction modes 2 to 33 have the horizontal directionality, andintra prediction modes 34 to 66 have the vertical directionality.Meanwhile, strictly speaking, intra prediction mode 34 may be consideredas being neither horizontal nor vertical, but it may be classified asbelonging to the horizontal directionality in terms of determining thetransform set of the secondary transform. This is because the input datais transposed to be used for the vertical direction mode symmetrical onthe basis of intra prediction mode 34, and the input data alignmentmethod for the horizontal mode is used for intra prediction mode 34.Transposing input data means that rows and columns of two-dimensionalblock data M×N are switched into N×M data. Intra prediction mode 18 andintra prediction mode 50 may represent a horizontal intra predictionmode and a vertical intra prediction mode, respectively, and intraprediction mode 2 may be called a right upward diagonal intra predictionmode because it has a left reference pixel and predicts in a rightupward direction. In the same manner, intra prediction mode 34 may becalled a right downward diagonal intra prediction mode, and intraprediction mode 66 may be called a left downward diagonal intraprediction mode.

In this case, mapping between the 35 transform sets and the intraprediction modes may be, for example, represented as in the followingtable. For reference, if an LM mode is applied to a target block, thesecondary transform may not be applied to the target block.

TABLE 2 intra mode 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 set 0 1 23 4 5 6 7 8 9 10 11 12 13 14 15 16 17 intra mode 18 19 20 21 22 23 24 2526 27 28 29 30 31 32 33 set 18 19 20 21 22 23 24 25 26 27 28 29 30 31 3233 intra mode 34 35 35 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 set34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 intra mode 52 5354 55 56 57 58 59 60 61 62 63 64 65 66 67 (LM) set 16 15 14 13 12 11 109 8 7 5 5 4 3 2 NULL

If a specific set is determined to be used, one of k transform kernelsin the specific set may be selected through the non-separable secondarytransform index. The encoding apparatus may derive a non-separablesecondary transform index indicating a specific transform kernel basedon the rate-distortion (RD) check, and may signal the non-separablesecondary transform index to the decoding apparatus. The decodingapparatus may select one from among k transform kernels in the specificset based on the non-separable secondary transform index. For example,the NSST index value 0 may indicate a first non-separable secondarytransform kernel, the NSST index value 1 may indicate a secondnon-separable secondary transform kernel, and the NSST index value 2 mayindicate a third non-separable secondary transform kernel.Alternatively, the NSST index value 0 may indicate that the firstnon-separable secondary transform is not applied to a target block, andthe NSST index values 1 to 3 may indicate the three transform kernels.

Referring back to FIG. 4, the transformer may perform the non-separablesecondary transform based on the selected transform kernels, and mayobtain modified (secondary) transform coefficients. As described above,the modified transform coefficients may be derived as transformcoefficients quantized through the quantizer, and may be encoded andsignaled to the decoding apparatus and transferred to thedequantizer/inverse transformer in the encoding apparatus.

As described above, if the secondary transform is omitted, (primary)transform coefficients, which are an output of the primary (separable)transform, may be derived as transform coefficients quantized throughthe quantizer as described above, and may be encoded and signaled to thedecoding apparatus and transferred to the dequantizer/inversetransformer in the encoding apparatus.

The inverse transformer may perform a series of procedures in theinverse order to that in which they have been performed in theabove-described transformer. The inverse transformer may receive(dequantized) transformer coefficients, and derive (primary) transformcoefficients by performing a secondary (inverse) transform (S450), andmay obtain a residual block (residual samples) by performing a primary(inverse) transform on the (primary) transform coefficients (S460). Inthis connection, the primary transform coefficients may be calledmodified transform coefficients from the viewpoint of the inversetransformer. As described above, the encoding apparatus and the decodingapparatus may generate the reconstructed block based on the residualblock and the predicted block, and may generate the reconstructedpicture based on the reconstructed block.

The decoding apparatus may further include a secondary inverse transformapplication determinator (or an element to determine whether to apply asecondary inverse transform) and a secondary inverse transformdeterminator (or an element to determine a secondary inverse transform).The secondary inverse transform application determinator may determinewhether to apply a secondary inverse transform. For example, thesecondary inverse transform may be an NSST or an RST, and the secondaryinverse transform application determinator may determine whether toapply the secondary inverse transform based on a secondary transformflag obtained by parsing the bitstream. In another example, thesecondary inverse transform application determinator may determinewhether to apply the secondary inverse transform based on a transformcoefficient of a residual block.

The secondary inverse transform determinator may determine a secondaryinverse transform. In this case, the secondary inverse transformdeterminator may determine the secondary inverse transform applied tothe current block based on an NSST (or RST) transform set specifiedaccording to an intra prediction mode. In an embodiment, a secondarytransform determination method may be determined depending on a primarytransform determination method. Various combinations of primarytransforms and secondary transforms may be determined according to theintra prediction mode. Further, in an example, the secondary inversetransform determinator may determine a region to which a secondaryinverse transform is applied based on the size of the current block.

Meanwhile, as described above, if the secondary (inverse) transform isomitted, (dequantized) transform coefficients may be received, theprimary (separable) inverse transform may be performed, and the residualblock (residual samples) may be obtained. As described above, theencoding apparatus and the decoding apparatus may generate thereconstructed block based on the residual block and the predicted block,and may generate the reconstructed picture based on the reconstructedblock.

In the present disclosure, a reduced secondary transform (RST) in whichthe size of a transform matrix (kernel) is reduced may be applied in theconcept of NSST in order to reduce the amount of computation and memoryrequired for the non-separable secondary transform.

The transform kernel, the transform matrix, and the coefficientconstituting the transform kernel matrix, that is, the kernelcoefficient or the matrix coefficient, described in the presentdisclosure may be expressed in 8 bits. This may be a condition forimplementation in the decoding apparatus and the encoding apparatus, andmay reduce the amount of memory required to store the transform kernelwith a performance degradation that can be reasonably accommodatedcompared to the existing 9 bits or 10 bits. In addition, the expressingof the kernel matrix in 8 bits may allow a small multiplier to be used,and may be more suitable for single instruction multiple data (SIMD)instructions used for optimal software implementation.

In the present specification, the term “RST” may mean a transform whichis performed on residual samples for a target block based on a transformmatrix whose size is reduced according to a reduced factor. In the caseof performing the reduced transform, the amount of computation requiredfor transform may be reduced due to a reduction in the size of thetransform matrix. That is, the RST may be used to address thecomputational complexity issue occurring at the non-separable transformor the transform of a block of a great size.

RST may be referred to as various terms, such as reduced transform,reduced secondary transform, reduction transform, simplified transform,simple transform, and the like, and the name which RST may be referredto as is not limited to the listed examples. Alternatively, since theRST is mainly performed in a low frequency region including a non-zerocoefficient in a transform block, it may be referred to as aLow-Frequency Non-Separable Transform (LFNST).

When the secondary inverse transform is performed based on RST, theinverse transformer 235 of the encoding apparatus 200 and the inversetransformer 322 of the decoding apparatus 300 may include an inversereduced secondary transformer which derives modified transformcoefficients based on the inverse RST of the transform coefficients, andan inverse primary transformer which derives residual samples for thetarget block based on the inverse primary transform of the modifiedtransform coefficients. The inverse primary transform refers to theinverse transform of the primary transform applied to the residual. Inthe present disclosure, deriving a transform coefficient based on atransform may refer to deriving a transform coefficient by applying thetransform.

FIG. 6 is a diagram illustrating an RST according to an embodiment ofthe present disclosure.

In the present specification, the term “target block” may mean a currentblock or a residual block on which coding is performed.

In the RST according to an example, an N-dimensional vector may bemapped to an R-dimensional vector located in another space, so that thereduced transform matrix may be determined, where R is less than N. Nmay mean the square of the length of a side of a block to which thetransform is applied, or the total number of transform coefficientscorresponding to a block to which the transform is applied, and thereduced factor may mean an R/N value. The reduced factor may be referredto as a reduced factor, reduction factor, simplified factor, simplefactor or other various terms. R may be referred to as a reducedcoefficient, but according to circumstances, the reduced factor may meanR. Further, according to circumstances, the reduced factor may mean theN/R value.

In an example, the reduced factor or the reduced coefficient may besignaled through a bitstream, but the example is not limited to this.For example, a predefined value for the reduced factor or the reducedcoefficient may be stored in each of the encoding apparatus 200 and thedecoding apparatus 300, and in this case, the reduced factor or thereduced coefficient may not be signaled separately.

The size of the reduced transform matrix according to an example may beR×N less than N×N, the size of a conventional transform matrix, and maybe defined as in Equation 4 below.

$\begin{matrix}{T_{RxN} = \begin{bmatrix}\begin{matrix}t_{11} \\t_{21}\end{matrix} & \begin{matrix}t_{12} \\t_{22}\end{matrix} & \begin{matrix}t_{13} \\t_{23}\end{matrix} & \ldots & \begin{matrix}t_{1N} \\t_{2N}\end{matrix} \\\; & \vdots & \; & \ddots & \vdots \\t_{R\; 1} & t_{R\; 2} & t_{R\; 3} & \ldots & t_{RN}\end{bmatrix}} & \left\lbrack {{Equation}\mspace{14mu} 4} \right\rbrack\end{matrix}$

The matrix T in the Reduced Transform block shown in FIG. 6A may meanthe matrix T_(R×N) of Equation 4. As shown in FIG. 6A, when the reducedtransform matrix T_(R×N) is multiplied to residual samples for thetarget block, transform coefficients for the target block may bederived.

In an example, if the size of the block to which the transform isapplied is 8×8 and R=16 (i.e., R/N=16/64=1/4), the RST according to FIG.6A may be expressed as a matrix operation as shown in Equation 5 below.In this case, memory and multiplication calculation can be reduced toapproximately 1/4 by the reduced factor.

In this document, matrix operation can be understood as an operation toobtain a column vector by multiplying the matrix and the column vectorby placing the matrix on the left side of the column vector.

$\begin{matrix}{\begin{bmatrix}\begin{matrix}t_{1,1} \\t_{2,1}\end{matrix} & \begin{matrix}t_{1,2} \\t_{2,2}\end{matrix} & \begin{matrix}t_{1,3} \\t_{2,3}\end{matrix} & \ldots & \begin{matrix}t_{1,64} \\t_{2,64}\end{matrix} \\\; & \vdots & \; & \ddots & \vdots \\t_{16,1} & t_{16,2} & t_{16,3} & \ldots & t_{16,64}\end{bmatrix} \times \begin{bmatrix}r_{1} \\r_{2} \\\vdots \\\vdots \\\vdots \\r_{64}\end{bmatrix}} & \left\lbrack {{Equation}\mspace{14mu} 5} \right\rbrack\end{matrix}$

In Equation 5, r₁ to r64 may represent residual samples for the targetblock and may be specifically transform coefficients generated byapplying a primary transform. As a result of the calculation of Equation5, transform coefficients c_(i) for the target block may be derived, anda process of deriving c_(i) may be as in Equation 6.

$\begin{matrix}{{{for}\mspace{14mu} i\mspace{14mu}{from}\mspace{14mu} 1\mspace{14mu}{to}\mspace{14mu}{\text{R}\text{:}}}{c_{i} = {0{for}\mspace{14mu} j\mspace{14mu}{from}\mspace{14mu} 1\mspace{14mu}{to}\mspace{14mu}{\text{N}\text{:}}}}{c_{i}+={t_{i,j}*r_{j}}}} & \left\lbrack {{Equation}\mspace{14mu} 6} \right\rbrack\end{matrix}$

As a result of the calculation of Equation 6, transform coefficients c₁to c_(R) for the target block may be derived. That is, when R=16,transform coefficients c₁ to C₁₆ for the target block may be derived.If, instead of RST, a regular transform is applied and a transformmatrix of 64×64 (N×N) size is multiplied to residual samples of 64×1(N×1) size, then only 16 (R) transform coefficients are derived for thetarget block because RST was applied, although 64 (N) transformcoefficients are derived for the target block. Since the total number oftransform coefficients for the target block is reduced from N to R, theamount of data transmitted by the encoding apparatus 200 to the decodingapparatus 300 decreases, so efficiency of transmission between theencoding apparatus 200 and the decoding apparatus 300 can be improved.

When considered from the viewpoint of the size of the transform matrix,the size of the regular transform matrix is 64×64 (N×N), but the size ofthe reduced transform matrix is reduced to 16×64 (R×N), so memory usagein a case of performing the RST can be reduced by an R/N ratio whencompared with a case of performing the regular transform. In addition,when compared to the number of multiplication calculations N×N in a caseof using the regular transform matrix, the use of the reduced transformmatrix can reduce the number of multiplication calculations by the R/Nratio (R×N).

In an example, the transformer 232 of the encoding apparatus 200 mayderive transform coefficients for the target block by performing theprimary transform and the RST-based secondary transform on residualsamples for the target block. These transform coefficients may betransferred to the inverse transformer of the decoding apparatus 300,and the inverse transformer 322 of the decoding apparatus 300 may derivethe modified transform coefficients based on the inverse reducedsecondary transform (RST) for the transform coefficients, and may deriveresidual samples for the target block based on the inverse primarytransform for the modified transform coefficients.

The size of the inverse RST matrix T_(N×R) according to an example isN×R less than the size N×N of the regular inverse transform matrix, andis in a transpose relationship with the reduced transform matrix T_(R×N)shown in Equation 4.

The matrix Tt in the Reduced Inv. Transform block shown in FIG. 6B maymean the inverse RST matrix T_(R×N) ^(T) (the superscript T meanstranspose). When the inverse RST matrix T_(R×N) ^(T) is multiplied tothe transform coefficients for the target block as shown in FIG. 6B, themodified transform coefficients for the target block or the residualsamples for the target block may be derived. The inverse RST matrixT_(R×N) ^(T) may be expressed as (T_(R×N) ^(T))_(N×R).

More specifically, when the inverse RST is applied as the secondaryinverse transform, the modified transform coefficients for the targetblock may be derived when the inverse RST matrix T_(R×N) ^(T) ismultiplied to the transform coefficients for the target block.Meanwhile, the inverse RST may be applied as the inverse primarytransform, and in this case, the residual samples for the target blockmay be derived when the inverse RST matrix T_(R×N) ^(T) is multiplied tothe transform coefficients for the target block.

In an example, if the size of the block to which the inverse transformis applied is 8×8 and R=16 (i.e., R/N=16/64=1/4), then the RST accordingto FIG. 6B may be expressed as a matrix operation as shown in Equation 7below.

$\begin{matrix}{\begin{bmatrix}\begin{matrix}t_{1,1} \\\begin{matrix}t_{1,2} \\t_{1,3}\end{matrix}\end{matrix} & \; & \begin{matrix}t_{2,1} \\\begin{matrix}t_{2,2} \\t_{2,3}\end{matrix}\end{matrix} & \ldots & \begin{matrix}t_{16,1} \\\begin{matrix}t_{16,1} \\t_{16,1}\end{matrix}\end{matrix} \\{\vdots\;} & \; & \vdots & \; & \vdots \\\; & \vdots & \; & \ddots & \vdots \\t_{16,1} & \; & t_{16,2} & \ldots & t_{16,64}\end{bmatrix} \times \begin{bmatrix}c_{1} \\c_{11} \\\vdots \\c_{16}\end{bmatrix}} & \left\lbrack {{Equation}\mspace{14mu} 7} \right\rbrack\end{matrix}$

In Equation 7, c₁ to C₁₆ may represent the transform coefficients forthe target block. As a result of the calculation of Equation 7, r_(j)representing the modified transform coefficients for the target block orthe residual samples for the target block may be derived, and theprocess of deriving r_(j) may be as in Equation 8.

$\begin{matrix}{{{for}\mspace{14mu} i\mspace{14mu}{from}\mspace{14mu} 1\mspace{14mu}{to}\mspace{14mu}{\text{N}\text{:}}}{r_{j} = 0}{{for}\mspace{14mu} j\mspace{14mu}{from}\mspace{14mu} 1\mspace{14mu}{to}\mspace{14mu}{\text{R}\text{:}}}{r_{j}+={t_{j,i}*c_{i}}}} & \left\lbrack {{Equation}\mspace{14mu} 8} \right\rbrack\end{matrix}$

As a result of the calculation of Equation 8, r₁ to r_(N) representingthe modified transform coefficients for the target block or the residualsamples for the target block may be derived. When considered from theviewpoint of the size of the inverse transform matrix, the size of theregular inverse transform matrix is 64×64 (N×N), but the size of thereduced inverse transform matrix is reduced to 64×16 (R×N), so memoryusage in a case of performing the inverse RST can be reduced by an R/Nratio when compared with a case of performing the regular inversetransform. In addition, when compared to the number of multiplicationcalculations N×N in a case of using the regular inverse transformmatrix, the use of the reduced inverse transform matrix can reduce thenumber of multiplication calculations by the R/N ratio (N×R).

A transform set configuration shown in Table 2 may also be applied to an8×8 RST. That is, the 8×8 RST may be applied according to a transformset in Table 2. Since one transform set includes two or three transforms(kernels) according to an intra prediction mode, it may be configured toselect one of up to four transforms including that in a case where nosecondary transform is applied. In a transform where no secondarytransform is applied, it may be considered to apply an identity matrix.Assuming that indexes 0, 1, 2, and 3 are respectively assigned to thefour transforms (e.g., index 0 may be allocated to a case where anidentity matrix is applied, that is, a case where no secondary transformis applied), an NSST index as a syntax element may be signaled for eachtransform coefficient block, thereby designating a transform to beapplied. That is, through the NSST index, it is possible to designate an8×8 NSST for atop-left 8×8 block and to designate an 8×8 RST in an RSTconfiguration. The 8×8 NSST and the 8×8 RST refer to transformsapplicable to an 8×8 region included in the transform coefficient blockwhen both W and H of the target block to be transformed are equal to orgreater than 8, and the 8×8 region may be a top-left 8×8 region in thetransform coefficient block. Similarly, a 4×4 NSST and a 4×4 RST referto transforms applicable to a4×4region included in the transformcoefficient block when both W and H of the target block to are equal toor greater than 4, and the 4×4regionmay bea top-left 4×4 region in thetransform coefficient block.

If the (forward) 8×8 RST illustrated in Equation 4 is applied, 16significant transform coefficients are generated. Thus, it is consideredthat 64 pieces of input data forming the 8×8 region is reduced to 16pieces of output data, and only 1/4 of the region is filled withsignificant transform coefficients from the perspective of atwo-dimensional region. Accordingly, the 16 pieces of output dataobtained by applying the forward 8×8 RST, for example, the top-leftregion (transform coefficients 1 to 16, that is, c₁, c₂, . . . , C₁₆obtained through Equation 6) of the block as shown in FIG. 7, may befilled in the diagonal direction scanning order from 1 to 16.

FIG. 7 is a diagram illustrating a transform coefficient scanning orderaccording to an embodiment of the present disclosure. As describedabove, when the forward scanning order starts from a first transformcoefficient, reverse scanning may be performed in directions and ordersindicated by arrows shown in FIG. 7 from 64th to 17th transformcoefficients in the forward scanning order.

In FIG. 7, a top-left 4×4 region is a region of interest (ROI) filledwith significant transform coefficients, and the remaining region isempty. The empty region may be filled with 0s by default.

That is, when an 8×8 RST with a 16×64 forward transform matrix isapplied to the 8×8 region, output transform coefficients may be arrangedin the top-left 4×4 region, and the region where no output transformcoefficient exists may be filled with 0s (from the 64th to 17thtransform coefficients) according to the scanning order of FIG. 7.

If a non-zero significant transform coefficient is found outside the ROIof FIG. 7, it is certain that the 8×8 RST has not been applied, and thusNSST index coding may be omitted. On the contrary, if a non-zerotransform coefficient is not found outside the ROI of FIG. 7 (e.g., if atransform coefficient is set to 0 in a region other than the ROI in acase where the 8×8 RST is applied), the 8×8 RST is likely to have beenapplied, and thus NSST index coding may be performed. This conditionalNSST index coding may be performed after a residual coding processbecause it is necessary to check the presence or absence of a non-zerotransform coefficient.

The present disclosure discloses methods for optimizing a design and anassociation of an RST that can be applied to a 4×4 block from an RSTstructure described in this embodiment. Some concepts can be applied notonly to a 4×4 RST but also to an 8×8 RST or other types of transforms.

FIG. 8 is a flowchart illustrating an inverse RST process according toan embodiment of the present disclosure.

Each operation disclosed in FIG. 8 may be performed by the decodingapparatus 300 illustrated in FIG. 3. Specifically, S800 may be performedby the dequantizer 321 illustrated in FIGS. 3, and S810 and S820 may beperformed by the inverse transformer 322 illustrated in FIG. 3.Therefore, a description of specific details overlapping with thoseexplained above with reference to FIG. 3 will be omitted or will be madebriefly. In the present disclosure, an RST may be applied to a transformin a forward direction, and an inverse RST may mean a transform appliedto an inverse direction.

In an embodiment, the specific operations according to the inverse RSTmay be different from the specific operations according to the RST onlyin that their operation orders are opposite to each other, and thespecific operations according to the inverse RST may be substantiallysimilar to the specific operations according to the RST. Accordingly, aperson skilled in the art will readily understand that the descriptionsof S800 to S820 for the inverse RST described below may be applied tothe RST in the same or similar manner.

The decoding apparatus 300 according to an embodiment may derive thetransform coefficients by performing dequantization on the quantizedtransform coefficients for the target block (S800).

The decoding apparatus 300 may determine whether to apply an inversesecondary transform before the inverse secondary transform. For example,the inverse secondary transform may be an NSST or an RST. For example,the decoding apparatus may determine whether to apply the inversesecondary transform based on a secondary transform flag parsed from abitstream. In another example, the decoding apparatus may determinewhether to apply the inverse secondary transform based on a transformcoefficient of a residual block.

The decoding apparatus 300 may determine an inverse secondary transform.In this case, the decoding apparatus 300 may determine the secondaryinverse transform applied to the current block based on an NSST (or RST)transform set specified according to an intra prediction mode. In anembodiment, a secondary transform determination method may be determineddepending on a primary transform determination method. For example, itmay be determined to apply an RST or LFNST only when DCT-2 is applied asa transform kernel in the primary transform. Alternatively, variouscombinations of primary transforms and secondary transforms may bedetermined according to the intra prediction mode.

Further, in an example, the decoding apparatus 300 may determine aregion to which the inverse secondary transform is applied based on thesize of the current block before determining the inverse secondarytransform.

The decoding apparatus 300 according to an embodiment may select atransform kernel (S810). More specifically, the decoding apparatus 300may select the transform kernel based on at least one of informations ona transform index, a width and height of a region to which the transformis applied, an intra prediction mode used in image decoding, and a colorcomponent of the target block. However, the example is not limited tothis, and for example, the transform kernel may be predefined, andseparate information for selecting the transform kernel may not besignaled.

In one example, information on the color component of the target blockmay be indicated through CIdx. If the target block is a luma block, CIdxmay indicate 0, and if the target block is a chroma block, for example,a Cb block or a Cr block, then CIdx may indicate a non-zero value (forexample, 1).

The decoding apparatus 300 according to an embodiment may apply theinverse RST to transform coefficients based on the selected transformkernel and the reduced factor (S820).

Hereinafter, a method for determining a secondary NSST set, that is, asecondary transform set or a transform set, in view of an intraprediction mode and the size of a block according to an embodiment ofthe present disclosure is proposed.

In an embodiment, a set for a current transform block may be configuredbased on the intra prediction mode described above, thereby applying atransform set including transform kernels having various sizes to thetransform block. Transform sets in Table 3 are expressed using 0 to 3 asin Table 4.

TABLE 3 Intra mode 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 NSST Set 0 02 2 2 2 2 2 2 2 2 2 2 18 18 18 18 Intra mode 17 18 19 20 21 22 23 24 2526 27 28 29 30 31 32 33 NSST Set 18 18 18 18 18 18 18 34 34 34 34 34 3434 34 34 34 Intra mode 34 35 36 37 38 39 40 41 42 43 44 45 45 47 48 4950 NSST Set 34 34 34 34 34 34 34 34 34 34 34 18 18 18 18 18 18 Intramode 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 NSST Set 18 18 1818 18 2 2 2 2 2 2 2 2 2 2 2

TABLE 4 Intra mode 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 NSST Set 0 01 1 1 1 1 1 1 1 1 1 1 2 2 2 2 Intra mode 17 18 19 20 21 22 23 24 25 2627 28 29 30 31 32 33 NSST Set 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 Intramode 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 NSST Set 3 3 3 33 3 3 3 3 3 3 2 2 2 2 2 2 Intra mode 51 52 53 54 55 56 57 58 59 60 61 6263 64 65 66 NSST Set 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1

Indexes 0, 2, 18, and 34 illustrated in Table 3 correspond to 0, 1, 2,and 3 in Table 4, respectively. In Table 3 and Table 4, only fourtransform sets are used instead of 35 transform sets, therebysignificantly reducing memory space.

Various numbers of transform kernel matrices that may be included ineach transform set may be set as shown in the following tables.

TABLE 5 0 NSST Set (DC, Planar) 1 2 3 # of transform 2 2 2 2 kernels

TABLE 6 0 NSST Set (DC, Planar) 1 2 3 # of transform 2 1 1 1 kernels

TABLE 7 0 NSST Set (DC, Planar) 1 2 3 # of transform 1 1 1 1 kernels

According to Table 5, two available transform kernels are used for eachtransform set, and accordingly a transform index ranges from 0 to 2.

According to Table 6, two available transform kernels are used fortransform set 0, that is, a transform set according to a DC mode and aplanar mode among intra prediction modes, and one transform kernel isused for each of the remaining transform sets. Here, an availabletransform index for transform set 1 ranges from 0 to 2, and a transformindex for the remaining transform sets 1 to 3 ranges from 0 to 1.

According to Table 7, one available transform kernel is used for eachtransform set, and accordingly a transform index ranges from 0 to 1.

In transform set mapping of Table 3, a total of four transform sets maybe used, and the four transform sets may be rearranged to bedistinguished by indexes 0, 1, 2, and 3 as shown in Table 4. Table 8 andTable 9 illustrate four transform sets available for secondarytransform, wherein Table 8 presents transform kernel matrices applicableto an 8×8 block, and Table 9 presents transform kernel matricesapplicable to a 4×4 block. Table 8 and Table 9 include two transformkernel matrices per transform set, and two transform kernel matrices maybe applied to all intra prediction modes as shown in Table 5.

TABLE 8 const int g_aiNsst8x8[4][2][16][64] = {  {//0  {  {−118,22,21,3,4,1,2,1,31,−17,−3,0,−1,0,0,0,16,0,−4,0,−1,0,0,0,2,0,0,0,0,0,0,0,3,0,−1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},  {−22,−88,46,0,9,0,3,0,−60,29,−7,3,0,1,0,0,35,2,−8,−1,−2,0,−1,0,−3,3,0,0,0,0,0,0,7,2,−2,−1,0,0,0,0,−1,1,0,0,0,0,0,0,3,1,−1,0,0,0,0,0,0,0,0,0,0,0,0,0},  {−10,68,−13,−7,−3,−2,−1,−1,−90,1,32,−1,4,0,2,0,32,−32,−3,4,−1,1,0,0,7,2,−5,0,0,0,0,0,6,−3,0,0,0,0,0,0,1,0,−1,0,

,0,0,0,2,− 1,0,0,0,0,0,0,1,0,0,0,

,0,0,0},  {16,−16,12,1,−1,0,0,0,−10,−111,20,18,4,4,2,1,20,24,−33,−1,0,0,0,0,−4,17,1,−7,0,−1,0,0,0,4,0,0,0,0,0,0,−1,3,0,−1,0,0,0,0,0,1,0,0,0,0,0,0,−1,1,0,0,0,0,0,

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 },  {  {70,−48,6,−4,1,−1,1,0,−75,43,2,1,0,0,0,0,33,−5,−12,3,−2,1,−1,0,−9,−9,9,0,0,0,0,0,3,6,−2,−2,0,0,0,0,−2,−2,0,1,0,0,0,0,2,1,0,0,0,0,0,0,−1,−1,0,0,0,0,0,0},  {−81,3,14,2,3,0,1,0,−13,55,−19,1,−3,1,−1,0,53,−46,−4,6,−2,1,−1,0,−18,3,20,−7,2,−1,1,0,7,6,−12,−1,1,−1,0,0,−2,−4,3,4,−1,0,0,0,3,1,−1,−2,0,0,0,0,−1,−1,1,0,0,0,0,0},  {48,−19,12,−3,1,−1,0,0,54,3,−29,2,−4,1,−2,0,−6,−62,34,2,3,0,1,0,−34,49,6,−15,4,−3,1,−1,7,−4,−23,9,−1,1,0,0,−5,−1,11,2,−3,1,0,0,1,0,−3,−3,1,0,0,0,−3,1,2,1,0,0,0,0},  {31,18,−23,7,−3,2,−1,1,50,−5,10,−7,−1,−2,0,−1,33,−20,−36,10,−2,2,−1,1,−2,−52,50,3,−2,1,0,0,−22,40,0,−27,7,−4,2,−1,−1,−6,−20,15,3,0,1,0,−3,3,9,2,−6,0,0,0,0,−3,−2,−2,0,1,0,0},  {−19,−75,28,1,5,0,2,0,37,42,−1,−14,0,−3,0,−1,−54,12,−29,18,−1,3,−1,1,35,−9,18,1,−7,2,−2,1,−10,−5,−2,−11,6,−1,1,0,2,5,0,6,0,−1,0,0,−2,−1,−2,−1,−2,0,0,0,1,0,2,1,0,1,0,0},  {17,4,16,−11,1,−2,1,−1,43,33,−28,12,−4,2,−2,1,42,7,−2,−15,−3,−2,−1,−1,11,−30,−45,20,1,3,0,1,4,−38,54,7,−7,2,−1,1,−11,23,−6,−25,7,−3,1,−1,0,−10,−6,14,2,0,1,0,−2,3,6,−4,−3,0,0,0},  {−17,−29,9,−13,4,−2,2,−1,−18,−30,−3,20,−3,3,−1,1,−14,−20,25,−17,5,−1,2,0,−21,10,12,23,−2,1,−1,0,4,17,34,−47,−4,−2,0,−1,−19,29,−57,7,26,−5,5,−1,10,−8,9,31,−18,−1,−1,0,−1,2,0,−14,−5,5,−1,1},  {−19,−64,−9,1,0,1,0,0,30,−30,47,−4,4,−1,2,0,42,27,−13,−20,1,−4,0,−1,−52,4,−20,12,4,1,1,0,16,10,−3,14,−9,2,−2,1,2,−17,12,−6,−3,2,−1,1,−1,9,2,−8,5,−1,0,0,−3,0,−5,3,2,−1,0,0},  {−10,−33,22,2,−2,−1,−1,0,−6,−41,−21,−1,4,2,1,1,29,2,44,13,−7,1,−2,0,25,−28,−1,−44,9,−4,3,−1,−48,26,0,30,12,−3,3,−1,10,20,−8,−10,−24,2,−4,1,−4,−13,−8,5,12,6,0,2,7,2,15,−3,−2,−4,−1,−1},  {5,5,6,−8,2,−1,1,0,15,20,0,0,2,−1,0,0,28,41,6,−13,3,−2,1,−1,15,24,−32,−15,1,−5,0,−2,−21,−7,−55,−7,11,0,3,0,−9,−5,−29,57,14,1,3,0,13,−8,25,23,−26,−2,−3,0,3,−10,7,−13,−10,6,−1,1},  {−3,−24,−13,3,1,1,0,0,−4,−45,2,9,−4,0,−1,0,38,−24,7,10,2,2,1,1,63,4,18,6,−13,−1,−4,0,14,−57,−7,−27,−1,2,0,1,−15,−10,28,13,18,−2,4,−1,−1,17,6,−11,−10,−5,0,−2,−6,1,−11,5,3,4,1,0},  {5,15,74,−17,7,−6,3,−2,−2,−34,−44,−13,6,−2,1,−1,11,37,−21,39,−7,3,−1,1,−33,−7,16,−4,−16,5,−4,1,30,−4,−2,−14,9,1,0,1,−4,−15,6,5,6,−5,2,−1,−1,9,3,−3,−4,0,1,−1,−3,1,−7,2,1,1,0,0},  {−9,−13,−18,−4,0,−1,0,0,−18,−25,−33,−6,1,−1,0,0,−28,−38,−36,−2,4,0,0,0,−35,−44,−30,13,9,3,2,1,−28,−31,−6,30,10,4,3,1,−8,−7,19,35,4,1,1,0,2,0,26,16,−11,−3,−2,−1,0,−2,13,−5,−9,1,−1,0},  {0,0,−3,6,−10,3,−2,1,−1,−2,1,−11,14,−4,2,−2,−1,−1,−6,11,−14,4,−2,2,4,2,8,−11,17,−3,2,−1,−1,3,−14,8,−27,1,−4,1,1,−18,8,−40,46,2,7,0,−15,30,−31,65,−2,−22,4,−5,10,−16,18,−9,−44,17,−3,2},  {0,4,−18,25,2,1,−1,0,−2,−7,25,−35,−4,0,1,0,9,1,−22,27,11,−3,1,−1,−18,17,10,−21,−23,5,−3,1,16,−33,3,1,34,−1,3,0,−30,51,−7,0,−20,−9,0,−2,11,−19,−30,18,3,12,−2,2,6,−9,40,−9,−7,−3,−2,1},  {−7,−18,−74,0,−9,−1,−3,0,5,18,−49,28,−2,6,−1,2,−4,46,32,32,−9,3,−2,1,−30,−9,17,−17,−15,2,−3,0,7,−11,4,−26,6,2,0,1,6,−7,5,−2,6,−4,2,−1,−7,0,−3,−1,−2,−1,2,−1,−3,4,−1,−1,0,1,0,0},  }  },  {//2  {  {117,−39,−4,−5,−1,−2,0,−1,23,−4,−2,0,−1,0,0,0,−21,6,1,1,0,0,0,0,−11,3,1,0,0,0,0,0,−5,2,0,0,0,0,0,0,−3,1,0,0,0,0,0,0,−2,1,0,0,0,0,0,0,−1,0,0,0,0,0,0,0},  {22,−3,−2,−1,−1,0,0,0,−116,29,5,3,1,1,1,0,−27,−1,3,0,1,0,0,0,24,−5,−1,0,0,0,0,0,12,−1,−1,0,0,0,0,0,5,−1,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0},  {−21,17,−4,2,−1,1,0,0,23,4,−3,0,−1,0,0,0,−118,19,7,2,2,1,1,0,−18,−4,3,0,1,0,0,0,19,−5,−1,0,0,0,0,0,7,−1,−1,0,0,0,0,0,2,0,0,0,0,0,0,0,2,−1,0,0,0,0,0,0},  {−34,−101,54,−11,9,−3,4,−1,−7,−23,10,−1,1,0,0,0,−9,28,−11,3,−2,1,−1,0,6,8,−3,0,−1,0,0,0,8,1,−2,0,0,0,0,0,2,1,−1,0,0,0,0,0,1,2,−1,0,0,0,0,0,1,1,0,0,0,0,0,0},  {6,2,−3,0,0,0,0,0,28,−13,1,−1,0,−1,0,0,−12,−14,3,−1,1,0,0,0,117,−16,−8,−1,−2,0,−1,0,28,6,−4,0,−1,0,0,0,−12,3,1,0,1,0,0,0,−5,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0},  {6,16,1,−1,0,0,0,0,−23,−102,36,−8,6,−2,2,−1,−14,−49,10,0,1,0,1,0,−12,24,−9,3,−1,1,0,0,3,17,−4,1,−1,0,0,0,4,5,−2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0},  {8,0,0,−1,0,0,0,0,6,2,−1,0,0,0,0,0,24,−7,1,−1,0,0,0,0,−25,−11,3,−1,1,0,0,0,116,−14,−8,−1,−2,0,−1,0,33,2,−4,0,−1,0,0,0,7,2,0,0,0,0,0,0,−1,0,0,0,0,0,0,0},  {−9,−30,18,−5,2,−1,1,−1,12,44,0,0,−1,0,0,0,−14,−99,24,−4,3,−1,1,0,−14,−31,−3,2,0,0,0,0,−8,31,−7,2,0,0,0,0,2,5,0,0,0,0,0,0,1,−2,0,−1,0,0,0,0,0,1,−1,0,0,0,0,0},  {−19,−51,−93,27,−6,5,−4,2,−8,−21,−50,15,−1,2,−2,1,−2,−8,12,−4,4,−1,1,−1,−1,3,16,−4,1,−1,1,0,3,8,9,−2,0,0,0,0,0,3,4,−1,−1,0,0,0,0,0,2,2,0,0,0,0,0,0,1,0,0,0,0,0,0},  {2,−1,−1,0,0,0,0,0,6,−3,−1,0,0,0,0,0,0,−2,0,0,0,0,0,0,20,−3,1,0,0,0,0,0,−33,−13,3,0,1,0,0,0,117,−10,−9,0,−2,0,−1,0,24,6,−3,0,−1,0,0,0,−5,0,1,0,0,0,0,0},  {0,5,1,0,0,0,0,0,5,29,−10,2,−1,0,−1,0,−2,2,−10,1,−1,0,−1,0,8,94,−19,1,−2,0,−2,0,16,72,1,0,0,0,0,0,14,−7,7,1,1,1,0,0,2,−16,2,−1,0,0,0,0,−1,−2,0,0,0,0,0,0},  {7,19,48,−1,2,−1,2,0,−9,−35,−87,24,−6,4,−4,2,−6,−22,−51,7,0,0,−1,0,0,−5,29,−15,4,−2,1,−1,1,2,18,−6,1,−1,0,0,0,2,7,2,−1,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0},  {−2,0,1,0,0,0,0,0,−2,1,1,0,0,0,0,0,−2,−1,0,0,0,0,0,0,−2,2,1,1,0,0,0,0,0,−3,1,0,0,0,0,0,16,0,−1,0,−1,0,0,0,−96,6,6,0,2,0,1,0,−82,0,6,−1,2,0,1,0},  {2,9,−1,1,1,0,0,0,−4,−17,1,0,−1,0,0,0,7,41,−1,1,0,0,0,0,−9,−69,3,−3,1,−1,0,0,0,91,−13,6,−1,2,0,0,13,9,7,1,0,1,0,0,−1,−25,3,−4,0,0,0,0−4,3,−2,−1,0,0,0,0},  {−3,−7,−27,5,2,0,−1,0,4,9,42,7,0,1,2,0,−7,−14,−80,17,−10,4,−3,1,−5,−18,−39,−23,1,−3,−1,−1,4,2,59,−16,3,−2,1,0,2,4,17,5,1,0,0,0,0,2,−3,3,0,0,0,0,0,0,−2,1,0,0,0,0},  {3,−2,24,69,−55,22,−7,4,2,1,15,26,−21,7,−1,1,0,2,−3,−56,39,−15,5,−3,−1,1,−13,−26,15,−5,0,−1,−1,−1,−6,1,−1,0,0,0,−1,1,−2,1,−1,0,0,−1,0,3,0,0,0,0,0,0,0,1,1,0,0,0},  },  {   {−72,45,

,5,−1,2,−1,1,77,−39,2,−3,0,−1,0,0,−37,9.5,1,1,0,0,0,11,2,−4,1,−1,0,0,0,−4,−2,1,0,0,0,0,0,2,1,0,0,0,0,0,0,−2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0},  {−87,11,10,0,2,0,1,0,−28,33,−11,3,−2,1,−1,0,67,−31,0,0,−1,0,0,0,−31,5,6,−1,1,0,1,0,9,4,−3,0,0,0,0,0,−2,−2,0,0,0,0,0,0,3,−1,0,0,0,0,0,0−1,0,1,0,0,0,0,0},  {19,−50,33,−8,5,−2,2,−1,77,29,−31,2,−4,0,−2,0,−2,−40,10,2,1,0,0,0,−40,26,5,−2,1,−1,1,0,18,−3,−6,2,−1,0,0,0,−9,0,3,0,0,0,0,0,3,0,−1,0,0,0,0,0,−3,0,1,0,0,0,0,0},  {−45,−54,29,−8,6,−2,2,−1,−17,44,−21,1,−1,0,−1,0,−50,19,−6,8,−2,2,−1,1,57,−19,2,−2,−1,0,0,0,−20,0,3,−1,1,0,0,0,7,2,0,0,0,0,0,0,−3,0,−1,0,0,0,0,0,2,0,0,0,0,0,0,0},  {−5,30,−8,1,−1,0,0,0,−45,14,−13,8,−3,2,−1,1,−65,−39,33,−7,5,−1,2,0,−4,52,−17,−2,0,−1,0,0,37,−30,−5,5,−2,1,−1,0,−20,6,8,−2,1,0,0,0,8,−1,−2,0,0,0,0,0,−3,1,0,0,0,0,0,0},  {−23,−40,10,−1,2,0,1,0,−16,−29,25,−8,4,−2,2,−1,−22,57,−25,5,−2,0,−1,0,−56,−7,4,4,−1,2,0,1,61,−9,0,−3,0,−1,0,0,−27,0,3,0,1,0,1,0,11,4,−1,0,−1,0,0,0,−5,−1,−1,0,0,0,0,0},  {−5,−8,−52,29,−8,6,−3,2,10,46,52,−28,5,−5,2,−1,−28,−36,−48,15,1,1,−1,1,−1,2,39,1,−3,1,0,0,2,17,−15,−9,3,−2,0,−1,0,−8,1,7,−2,1,0,0,1,2,0,−4,1,0,0,0,0,−3,0,2,0,0,0,0},  {−2,10,13,−7,1,−1,1,0,−23,−9,−16,7,−1,1,−1,0,−47,−8,10,−3,0,0,0,0,−70,−17,13,0,4,−1,1,0,−36,52,−4,1,−1,0,0,0,51,−15,−10,1,−2,1,−1,0,−22,−4,5,0,1,0,0,0,3,1,0,0,0,0,0,0},  {−23,−69,−31,9,−2,2,−2,0,−6,−35,41,−4,0,0,1,0,8,−13,30,−17,7,−3,2,−1,−2,44,−35,4,0,−1,0,0,−34,−12,2,8,−2,2,0,1,30,6,1,−2,0,0,0,0,−12,−4,1,0,1,0,0,0,3,3,−1,0,0,0,0,0},  {1,5,−53,8,−3,2,−3,1,19,69,−2,17,−6,4,−1,1,−3,46,37,−20,2,−2,1,−1,−28,−35,−39,1,2,−1,0,0,0,−15,12,8,−2,2,0,0,5,4,4,−4,1,−1,0,0,−5,0,1,1,0,0,0,0,−1,1,−2,0,0,0,0,0},  {6,23,19,−1,1,0,1,0,1,19,−2,−12,5,−2,1,−1,1,25,−45,24,−5,2,−2,1,−27,38,4,−3,−7,3,−2,1,−35,−75,20,−3,5,−1,1,0,−27,19.3,−2,1,−1,1,0,−20,−1,−1,0,−1,0,0,0,1,4,−2,0,0,0,0,0},  {−3,−2,0,0,−1,0,0,0,−6,−5,0,−1,0,0,0,0,−10,−5,−1,−1,0,0,0,0,−28,−7,1,0,0,0,0,0,−80,−7,2,−1,1,0,0,0,−90,8,8,1,1,0,1,0,23,11,−3,−1,−1,0,0,0,2,−3,−1,1,0,0,0,0},  {0,−3,−8,9,−4,1,−2,0,−2,−5,−8,−8,5,0,2,0,6,19,26,33,−13,5,−2,1,−1,1,20,−70,12,−6,3,−2,−5,−8,−60,43,13,−2,2,0,5,1,35,5,−23,5,−2,1,0,−1,−8,−15,13,0,0,0,−1,1,0,7,−2,−2,1,0},  {−3,0,−19,−62,50,−15,−4,3,1,25,61,−45,10,−1,2,−2,−1,−26,−20,10,5,−4,1,4,7,24,−12,14,−12,5,−2,−4,−7,−15,10,−11,5,−1,0,2,2,5,0,2,1,−1,0,1,0,−1,−1,0,−1,0,0,−1,0,0,1,1,0,0,0},  {0,−2,−3,0,1,0,0,0,−1,−2,4,−2,0,1,0,0,−4,−8,4,8,−2,1,−1,0,−4,−14,9,−14,5,−1,1,0,1,8,14,6,5,−1,1,−1,2,83,−36,−1,−9,0,−2,0,38,−74,0,5,1,2,−1,1,−17,16,14,−2,1,−1,1,0},  {7,26,72,0,01,2,4,0,7,25,83,−5,3,1,4,0,−2,−4,10,−18,6,−4,0,−1,−2,−8,−37,−5,3,−3,0,−1,−2,8,−21,15,−1,2,0,0,0,2,6,2,−3,1,−1,0,3,−5,−2,−3,2,0,0,0,0,0,−1,1,0,0,0,0},  }  },  {//3  {  {−112,43,9,3,2,1,1,0,12,−31,10,0,2,0,1,0,23,−12,−8,3,−1,1,0,0,4,3,−4,−1,0,0,0,0,4,0,−1,−1,0,0,0,0,1.1,−1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},  {−21,−43,19,1,3,0,1,0,91,−3,−38,2,−5,1,−2,0,−28,51,2,−9,0,−2,0,−1,−7,−9,15,1,0,0,0,0,−3,2,−2,2,0,0,0,0,−1,−2,2,0,0,0,0,0,−1,1,0,0,0,0,0,0,−1,−1,1,0,0,0,0,0},  {23,−31,17,−6,4,−2,1,−1,−3,−78,10,16,0,4,0,2,34,13,−64,7,−1,1,−1,0,−13,42,1,−23,2,−3,0,−1,2,−1,1,2,0,−4,0,−1,0,−3,6,0,1,1,−1,0,0,0,0,1,0,0,1,0,0,−1,2,0,0,0,0,0,0},  {−40,−93,33,8,5,2,2,1,−53,18,−22,5,2,1,0,0,13,−17,9,−4,1,0,0,0,13,−5,−16,6,−1,1,0,1,10,−6,−6,2,−1,0,0,3,0,0,−2,−2,1,0,0,1,2,0,−1,0,0,0,0,1,0,0,0,0,0,0,0},  {−10,19,−3,−2,−1,0,0,0,−52,2,−38,9,1,2,0,1,13,56,−7,−45,4,−7,1,−3,1,−11,68,−10,−12,1,−3,0,9,−17,1,27,−8,2,−1,1,−3,1,−2.5,3,−2,1,−1,2,−3,−1,2,1,0,0,0,−1,0,1,0,1,0,1,0},  {12,20,61,−17,3,−4,1,−2,−21,−52,9,26,−1,4,0,2,−56,1,4,0,4,1,1,0,16,−47,−3,20,−2,2,0,1,−1,9,−37,−3,8,0,1,0,3,0,4,−12,−3,1,0,0,1,3,−5,2,−1,−1,0,0,1,0,1,−2,0,0,0,0},  {−17,−3,−6,0,4,0,1,0,−8,23,7,13,−4,3,−2,1,−75,−19,−44,6,20,−1,5,0,27,8,3,−61,−2,1,−1,0,4,−2,33,12,−27,−1,−3.0,4,1,−7,13,7,−5,0,−1,−1,−1,4,−2,1,2,1,0,2,−1,−2,1,0,−1,1,0},  {6,−24,18,−17,2,−3,2,−1,24,−11,56,−21,−6,−2,0,−1,18,−32,28,−6,−8,0,−1,1,26,−13,44,2,−25,3,−5,1,4,−15,4,56,−20,−1,−2,1,−6,8,−25,21,13,−5,1,−1,−2,−1,−3,1,4,1,1,0,−1,1,−5,3,2,−2,2,0},  {10,29,86,−25,1,−4,2,−2,13,62,−2,27,−9,2,−2,1,37,−4,−19,9,9,−1,1,0,−16,12,0,−10,3,1,1,0,−4,−5,10,−1,−2,−1,0,0,−1,−4,1,4,0,0,0,0,−1,−1,0,0,0,0,0,0,−1,−1,0,1,0,0,0,0},  {20,18,1,12,−5,4,−2,1,25,−17,−63,16,14,−1,2,0,20,−57,0,−48,1,0,0,1,−1,49,10,−7,−3,−23,5,−5,1,−12,26,2,5,−3,−3,1,−1,4,−1,4,1,1,0,0,0,−3,3,0,−1,0,0,1,0,1,0,1,0,−1,0,0,0},  {9,12,22,13,−4,−2,1,0,−9,−25,−25,−27,6,2,0,0,12,15,35,14,−11,−4,−1,−1,−12,−32,−29,−26,17,10,3,3,21,15,36,−13,−53,−6,−6,−1,−5,0,−2,50,16,−11,−3,−3,1,1,−4,−13,5,9,2,0,−1,−2,1,9,−1,−5,1,0},  {3,−11,−31,5,9,−1,0,0,15,9,10,25,−8,5,−1,0,38,−10,−24,7,41,−3,7,−1,−14,−71,0,−34,2,20,0,6,6,9,−45,−2,−26,−2,−1,0,−3,10,9,−14,−1,−5,−1,−3,6,1,−2,4,−2,1,1,−1,−2,3,2,4,1,0,1,0},  {−1,4,−3,14,1,−3,0,−1,−4,4,−11,9,3,2,−3,1,−1,7,−7,22,3,−7,0,−3,−5,8,−11,12,7,1,−1,0,−7,20,−15,43,13,−23,4,−6,−3,−6,−15,−19,85,−5,−5,0,2,−12,3,−41,4,39,−8,5,−2,0,−5,−4,−9,11,3,−2},  {−2,−4,6,19,−5,5,−1,2,1,−12,−24,11,16,−3,4,−1,−22,−51,6,−16,6,7,−1,2,−82,−2,5,1,−6,3,2,0,10,−56,−2.5,1,−1,1,1,−1,5,−33,0,−1,0,−1,0,1,0,0,−12,1,−2,1,−1,0,1,−2,−1,−3,0,0,0},  {5,2,1,23,1,1,0,−1,4,−4,−11,17,16,3,−3,0,9,−4,5,29,3,−8,4,−7,33,−1,21,2,71,−1,−1,0,1,−14,−2,3,8,63,−2,10,−10,2,−39,0,−12,10,23,−1,−2,−6,−2,−21,2,−7,7,2,0,−1,−6,−5,−4,2,−1,2},  {4,2,18,86,−4,5,−4,4,4,6,47,33,38,−9,6,−1,1,29,16,−22,24,16,−1,3,11,14,−9,1,−19,14,3,2,5,−2,5,−5,−3,−9,5,−1,−2,1,3,2,−3,−1,−2,1,−1,−1,1,3,−1,0,0,0,0,0,0,2,0,−1,1,0},  },  {  {102,−43,−2,−4,−1,−1,0,−1,35,10,−12,1,−3,0,−1,0,−33,32,−1,0,0,0,0,0,−17,2,8,−1,1,0,0,0,−6,−1,3,1,0,0,0,0,−3,−1,1.1,0,0,0,0,−2,0,0,0,0,0,0,0,−1,0,1,0,0,0,0,0},  {−51,4,9,0,2,0,1,0,100,−31,−11,−1,−3,0,−1,0,−5,40,−9,0,−2,0,−1,0,−26,5,12,−1,2,0,1,0,0,−6,2,2,0,0,0,0,−3,0,0,0,0,0,0,0,0,−1,0,0,0,0,0,0,−2,0,0,0,0,0,0},  {−26,5,3,2,0,1,0,0,7,41,−1,1,0,−2,0,−1,0,−88,2,33,−2,6,−1,2,0,30,−58,2,6,0,2,0,1,14,5,−18,2,−1,0,−1,0,3,2,−1,−3,1,−1,0,0,1,1,−1,−1,0,0,0,0,2,−1,0,0,0,0,0,0  {−13,−72,21,−1,4,0,2,0,27,−14,−7,0,−1,0,0,0,7,−40,0,6,0,2,0,1,49,9,−47,3,−5,1,−2,0,−15,55,−1,−14,0,−3,0,−1,−2,−1,14,−1,−1,0,0,0,−3,5,0,0,0,0,0,0,0,0,3,0,−1,0,0,0},  {34,81,−28,2,−6,1,−2,0,42,−11,21,−7,3,−2,1,−1,−22,−47,5,9,0,2,0,1,15,18,−30,−1,−2,0,−1,0,−13,21,13,−10,0,−2,0,−1,−1,−7,10,5,−1,0,0,0,−3,−1,−1,2,1,0,0,0,0,−1,0,0,0,0,0,0},  {23,9,8,−5,1,−1,1,−1,−4,−74,18,4,3,1,1,1,7,−6,−4,2,−1,1,0,0,−1,−50,−3,14,−1,4,0,1,28,8,−61,3,−2,1,−1,0,−10,39,−2,−24,2,−4,1,−1,4,1,13,−4,−4,0,−1,0,−2,4,2,1,−1,0,0,0},  {−9,−6,13,−2,3,−1,1,0,−35,−63,13,8,1,2,0,1,−72,13,−45,6,−1,2,−1,1,19,34,4,−23,3,−5,1,−2,2,−11,29,1,1,0,0,7,−9,−4,8,−2,1,0,0,0,1,−4,1,1,0,0,0,2,1,−2,−1,1,0,0,0},  {4,32,6,−3,−2,0,−1,0,−24,−11,−39,6,−2,2,−1,1,11,65,2,−14,−2,−3,0,−1,12,−10,−9,4,1,0,0,0,−1,60,5,−27,−1,−5,0,−2,−12,−17,52,7,−2,0,0,0,4,−11,−10,11,1,1,0,1,−3,0,6,−1,1,−1,0,0},  {11,14,−6,−3,0,−1,0,0,14,0,−5,2,−3,1,−1,0,22,19,−9,−17,1,−4,0,−1,92,7,51,−1,−8,0,−2,0,−29,−3,−8,41,3,5,1,2,−8,1,−21,−12,9,−1,2,0,−3,0,5,−5,−3,0,−1,0,1,1,−1,0,0,0,0,0},  {13,−1,32,−4,1,−2,1,−1,4,−35,10,1,3,−1,1,0,17,−14,38,−3,0,−1.0,0,5,−43,13,14,1,1,0,1,3,−16,30,9,−4,2,−1,1,1,−45,−11,52,−2,7,−1,3,−3,6,−56,9,12,0,3,0,−3,7,−8,−9,5,−1,1,0},  {7,1,−5,−1,−2,0,−1,0,13,15,−3,−6,−1,−1,0,−1,16,5,−12,3,−3,0,−1,0,32,2,−2,−22,4,−6,1,−2,94,−1,32,−7,−12.0,−4,0,−7,44,1,35,−6,3,−1,1,−16,9,4.5,12,−1,3,0,−4,−6,0,1,3,1,0,0},  {13,31,99,−18,11,−5,6,−2,9,33,−9,27,−6,6,−2,2,4,−21,−44,5,4,1,1,0,−6,−8,1,−16,2,−1,0,0,−5,−2,−6,−1,−2,1,0,0,−2,3,1,−7,−1,−1,0,−1,0,−1,4,1,−2,0,−1,0,0,0,0,0,0,0,0,0},  {−2,−10,−5,−1,2,−2,0,−1,6,4,20,−,1−4,0,−1,0,−1,−12,−5,0,2,−2,1,−1,4,0,25,−3,−6,−1,−2,0,−1,−12,−11,2,2,−3,1,−1,24,7,51,−9,−17,0,−5,0,−1,−12,0,87,−3,2,0,1,−13,6,−45,0,32,−2,5,0},  {7,14,22,0,−2,1,0,0,−5,−21,−45,−2,1,−1,0,−1,4,13,41,−7,−2,0,0,0,18,13,−29,12,2,1,1,1,−24,−47,2,−28,−2,−3,0,−1,50,31,−9,5,−12,−1,−3,0,−22,25,26,9,5,−1,1,0,4,−21,−4,7,4,1,1,1},  {−1,−2,−9,−3,0,−1,1,−1,4,17,63,−3,2,0,1,0,17,49,−22,44,−1,6,0,2,24,−40,−38,−18,18,−1,4,0,−31,−22,2,−28,−10,2,−2,1,2,−3,−4,5,−9,−4,−1,−1,−2,7,0,0,2,−1,−1,0,2,−2,0,1,−1,1,0,0},   {2,−3,−4

,22,−4,5,−2,2,5,−9,−64,26,−2,5,−2,2,10,−31,−48,−5,13,−2,3,−1,1,−48,−1,−36,14,−3,2,−1,−13,−23,6,−19,−4,3,−2,1,−1,−12,−2,−2,−9,3,−2,0,2,2,−9,3,−6,1,−1,0,1,5,−2,−1,−1,−1,0,−1},  }  } },

indicates data missing or illegible when filed

TABLE 9 const int g_aiNsst4x4[4][2][16][16] = {  { //0   {    {108,−44,−16,2,−43,19,6,−1,−11,6,2,−1,0,−1,0,0 },    {37,100,−55,−13,2,−26,14,2,−14,−22,13,4,1,2,−2,0 },    {28,−21,−8,6,102,−17,−31,1,−53,19,14,−3,−8,1,4,0 },    {−33,−38,−94,53,−5,−15,29,−8,7,19,19,−14,3,1,−4,1 },    {8,−11,27,−7,−15,−105,35,25,−5,37,−26,−5,4,17,−6,−6 },    {−25,1,14,−2,−36,12,15,−2,−98,3,29,1,55,−9,−20,2 },    {7,7,14,2,37,33,98,−37,−6,12,−42,10,−9,−15,−14,9 },    {0,33,−7,−2,−12,21,−26,1,3,100,−23,−27,8,−45,22,10 },    {16,28,39,108,−5,−8,−15,−22,−5,−17,−16,−27,4,5,3,4 },    {−10,−10,−34,0,−8,1,−25,6,−33,−26,−98,32,14,5,37,−16 },    {−16,3,5,4,−26,5,13,11,−47,1,13,−7,−104,3,42,−4 },    {3,5,1,17,16,33,26,109,3,−9,−9,−34,10,−2,−9,−26 },    {−3,14,−5,−3,30,−10,−11,−6,40,−11,−5,−7,108,−32,−22 },    {4,9,11,33,1,5,8,15,11,31,31,99,7,−6,20,−50 },    {3,−2,8,−11,10,4,28,−15,9,−4,23,−34,42,33,101,−16 },    {0,−2,−1,−13,−1,−7,−4,−35,−1,−8,−3,−38,−8,−31,−21,−109 },   },      {     { −118,32,21,3,27,4,−5,2,16,−3,−6,0,4,−2,0,0 },      {−30,−97,33,15,−51,3,25,3,21,30,−6,−8,6,3,−5,−2 },      {0,65,20,−16,−99,3,35,1,10,−15,−5,4,14,−1,−8,0 },      {24,4,63,−10,21,90,−5,−27,14,−8,−40,−1,−10,−18,0,8 },      {18,5,91,−2,25,−74,10,17,−21,−12,−20,4,−1,13,−3,−4 },      {8,−3,−1,−7,−24,−27,−90,8,69,−9,−38,10,−2,3,20,3 },      {19,30,−9,5,29,−15,43,5,69,75,−19,−29,−1,−5,−22,−5 },      {−3,21,22,−3,−20,9,−57,−9,−55,86,15,−22,15,10,7,−4 },      {−7,−17,−1,−112,7,−16,9,−41,8,15,17,26,−2,3,−1,9 },      {10,5,37,0,7,22,−11,36,44,−4,101,10,−8,15,10,−19 },      {2,6,11,34,−10,−29,1,−85,12,−7,35,−30,−46,−27,27,32 },      {8,−2,2,17,11,0,−4,−56,17,−21,9,−15,86,63,−30,−5 },      {−4,4,−8,−8,−4,12,16,−2,−2,−1,−22,−22,−54,90,53,−30 },      {−4,−7,−1,−24,−7,−2,−24,1,−4,−29,5,−75,−34,−2,−78,−37 },      {1,−6,0,−30,6,−1,11,25,0,−18,1,−79,46,−36,70,0 },      {1,0,−1,9,0,−7,3,−34,−1,3,−5,26,8,−38,20,−112 },      }     },  { //1   {   { −110,40,5,3,44,13,−12,−1,8,−15,−6,2,3,−2,4,2 },    {−47,−29,16,−1,−91,42,22,−2,20,40,−15,−5,10,−5,−13,1 },    {36,21,−3,−4,17,74,−32,−6,58,−17,−49,5,1,−39,−2,11 },    {−13,−93,27,2,49,−14,33,−5,51,−12,−1,−10,−1,−17,2,−2 },    {11,28,−3,−2,47,3,38,−10,1,85,−12,−27,17,1,−58,−5 },    {0,−35,34,−4,25,58,−4,−16,−83,2,−30,1,−36,23,2,12 },    {−17,−47,−93,16,9,−5,−42,30,−13,28,−25,18,0,−6,−12,14 },    {−5,4,20,−6,−15,−58,3,−24,−31,−16,−68,3,25,−59,−17,38 },    {6,−16,34,−4,9,−1,−49,−4,3,12,−7,24,95,51,4,2 },    {−3,2,38,−15,−7,−41,−58,−5,37,32,−20,17,−71,31,−21,4 },    {−3,−5,−28,−9,−13,−1,10,−18,9,−64,−30,−38,2,58,−68,−19 },    {5,16,−19,29,2,−10,49,−9,22,10,−53,8,−7,59,62,40 },    {−6,−8,−36,−86,1,1,−9,−75,4,14,14,−16,2,1,33,10 },    {0,4,−3,−55,5,9,41,30,4,−14,13,85,−2,10,−36,39 },    {1,−2,−6,63,−3,9,−9,−70,7,−5,49,20,−4,−1,−32,58 },    {1,0,11,−22,0,4,−15,51,3,−3,23,−70,3,8,−2,86 },   },      {      {−88,55,−6,3,66,−28,−8,1,−11,−10,11,−1,3,6,−1,−2 },      {−58,−19,26,−2,−28,75,−30,0,46,−43,−10,11,−7,−3,19,−5 },      {45,−34,29,−5,59,−1,−34,2,−7,−58,32,3,−26,32,7,−11 },      {−34,−72,43,−1,32,16,15,−18,−55,43,−31,7,19,−4,−5,9 },      {19,−3,−36,21,50,6,36,−22,30,−19,−63,16,−7,−42,51,6 },      {30,49,11,−9,2,32,−59,5,−52,1,−14,30,52,−32,27,−9 },      {9,18,77,−44,8,−42,−18,5,54,33,−24,11,−20,−29,2,1 },      {−21,−37,6,5,−32,−63,7,−1,−3,−26,34,23,37,−20,61,−40 },      {5,−26−16,4,36,13,−11,24,48,12,38,−36,67,−51,−32,5 },      {14,27,43,43,−1,2,29,−73,24,16,7,22,49,29,−31,−6 },      {0,12,27,−49,−13,−6,46,11,−26,−66,−12,−40,27,−13,0,55 },      {9,24,18,−31,19,46,59,17,5,52,28,−25,8,26,46,−54 },      {1,14,33,50,−7,12,3,−29,−25,5,46,−42,−50,−65,18,12 },      {−3,−2,−22,−59,4,16,28,−32,−11,−12,34,61,−24,−47,−43,−22 },      {−1,−4,−20,−31,5,6,−16,−44,17,33,44,14,7,25,50,77 },      {3,7,23,41,5,10,36,73,2,3,25,69,−3,−2,1,43 },      }     },      { //2      {        { −112,48,−1,3,−28,11,1,0,19,−8,0,0,10,−4,0,0 },        {−24,8,−2,1,112,−42,−3,0,31,−11,−3,1,−16,6,0,0 },        {37,87,−73,14,10,26,−20,3,−11,−17,14,−3,−6,−8,6,−1 },        {28,−6,−3,1,−19,9,−3,1,109,−38,−5,0,39,−13,−3,1 },        {9,18,−14,5,−32,−89,65,−9,−10,−38,27,−4,4,10,−8,2 },        {−22,−58,−66,78,−8,−16,−21,23,5,16,13,−17,4,6,6,−8 },        {2,−2,0,0,−26,6,1,1,36,−8,0,−1,−116,28,8,−1 },        {11,30,−16,−1,−10,−24,15,−3,28,89,−51,4,13,46,−25,−1 },        {2,8,8,−12,−21,−51,−68,61,−14,−30,−46,40,0,3,1,−3 },        {16,33,63,90,7,15,26,30,−5,−11,−25,−10,−4,−6,−13,−5 },        {−1,−1,−16,3,−4,−19,24,−10,6,28,−40,17,−23,−103,35,5 },        {−10,−24,−35,9,8,28,28,−30,−17,−44,−58,46,−4,11,−55,30 },        {−2,−6,−13,−17,12,26,53,83,8,18,37,60,1,0,−1,1 },        {1,4,7,−5,−4,−13,−26,16,6,20,36,−24,−16,−43,−91,49 },        {4,10,21,37,−5,−12,−27,−46,8,19,41,77,5,10,24,51 },        {−1,−2,−4,−9,2,5,13,29,−3,−7,−21,−43,7,19,47,102 },       },      {     { −98,41,−1,3,66,−21,−4,0,−14,−2,5,−1,1,3,−1,0 },      {59,36,−29,4,36,−61,21,−1,−61,32,3,−4,22,−3,−8,3 },      {−8,75,−46,6,−56,−33,24,2,58,−5,−2,−5,−18,3,−1,2 },      {47,4,4,−4,73,−11,−15,4,5,1,−38,12,−2,−61,22,3,−2 },      {−14,−50,−14,20,20,−21,56,−26,31,55,−52,8,−31,−23,14,5 },      {1,14,74,−51,−14,−49,−34,30,20,50,−9,−3,−12,−18,8,−1 },      {18,16,−3,2,38,21,−10,0,61,3,−15,5,89,−43,10,−2 },      {14,53,−7,2,6,61,−24,−3,−81,19,−36,18,−46,−47,41,−7 },      {−8,−22,−51,−18,9,22,−10,52,7,37,48,−54,−19,−41,−23,19 },      {−3,−4,−22,−81,8,13,47,51,−11,−29,−47,0,11,24,19,−8 },      {−8,30,−16,5,−13,−53,−12,8,−13,−59,14,4,−4,−57,58,−18 },      {5,21,50,−18,2,13,65,−26,−8,−36,16,−18,−11,−58,−42,30 },      {4,7,35,49,−2,−3,−1,25,−7,−16,−42,−85,7,18,34,44 },      {2,11,24,0,4,20,50,−7,5,27,65,−20,7,22,72,−39 },      {2,5,34,59,2,3,26,77,−2,−4,−9,20,−5,−12,−35,−61 },      {0,1,7,19,1,2,16,45,2,4,27,68,2,7,27,86 },      }     },      { //3      {        { 114,−38,−3,−2,20,23,−12,1,−22,18,4,−2,−5,−1,5,0 },       { 19,43,−17,2,−84,59,14,−6,−18,−34,33,−1,11,−20,−1,7 },        {−34,29,−21,2,45,55,−33,−7,−44,58,27,−17,−8,−21,32,0 },        {31,79,−42,3,56,−5,31,−17,42,−19,7,9,−10,20,−9,6 },        {15,29,−31,6,−49,−5,−39,12,25,49,−61,−15,−12,41,21,−33 },        {11,43,20,−7,−7,−45,46,9,−85,−18,−31,24,−22,−10,2,−8 },        {−10,−18,−78,37,12,−12,−37,9,−48,−46,−21,4,7,−3,−43,−10 },        {18,16,−15,7,−8,−71,−21,−2,6,−6,27,−45,40,−55,51,−1 },        {8,35,43,−12,26,30,−27,42,4,−21,−46,14,71,−26,−14,−21 },        {8,28,40,17,1,−10,−56,51,5,−31,20,−13,−77,−6,−8,18 },        {−5,−12,6,−24,17,19,16,−12,−11,−57,−7,−39,−27,11,43,−83 },        {−3,−20,−24,44,2,15,29,45,28,1,−12,62,−21,−47,53,−13 },        {1,2,17,74,4,0,33,47,−12,13,38,−31,33,55,−3,−21 },        {−3,−2,−19,−52,0,−12,−22,36,−20,−22,30,46,23,61,52,23 },        {2,6,33,60,5,9,−20,−63,−14,−32,−35,12,7,24,50,40 },        {5,11,20,16,−8,−20,−39,−42,4,13,49,65,4,1,−19,−68 },       },      {      { −98,30,5,8,67,−29,−4,−1,−18,14,1,−1,5,−2,0,0 },       {−15,−88,29,3,36,82,−38,−4,−18,−4,24,8,5,−6,−5,2 },       {63,24,−26,0,57,−3,5,2,−81,16,14,1,26,−14,−6,2 },       {−29,24,−39,13,−17,54,35,−18,−24,−77,19,18,20,11,−29,−4 },       {1,41,73,−34,−26,−6,−50,36,−24,−39,20,−15,25,13,−12,0 },       {−30,35,0,0,−63,−28,16,−3,−49,27,−19,−1,68,−29,−1,−2 },       {11,−32,22,−15,31,−18,31,16,−10,−41,−78,13,12,54,12,−50 },       {12,−29,−23,32,8,−64,−33,−23,19,−19,40,−11,29,36,−37,12 },       {−10,−24,−37,−82,3,−2,14,63,6,6,21,−28,5,8,−14,0 },       {−7,19,−43,18,−8,34,−51,2,−5,8,−33,−42,27,40,72,1 },       {−13,−17,−14,6,−33,−26,−14,1,−76,−10,5,−5,−90,20,9,−5 },       {−3,5,−13,−47,−11,0,−29,−30,0,35,22,36,10,24,19,−30 },       {−3,−2,87,−18,−5,8,60,−24,−9,22,41,−1,5,54,42,59 },       {5,−17,−10,−24,11,−29,−7,−23,8,−67,14,8,5,−57,68,22 },       {−2,6,−10,−48,−3,9,−34,−58,−7,3,−57,2,−6,0,−40,66 },       {−1,0,−12,35,−1,0,−18,68,−1,−2,−11,74,2,4,6,66 },      }      }     },

All of the illustrative transform kernel matrices shown in Table 8 aretransform kernel matrices multiplied by 128 as a scaling value. In ag_aiNsst8×8[N1][N2][16][64] array present in matrix arrays of Table 8,N1 denotes the number of transform sets (N1 is 4 or 35, distinguished byindex 0, 1, . . . , and N1-1), N2 denotes the number (1 or 2) oftransform kernel matrices included in each transform set, and [16][64]denotes a 16×64 reduced secondary transform (RST).

As shown in Table 3 and Table 4, when a transform set includes onetransform kernel matrix, either a first transform kernel matrix or asecond transform kernel matrix may be used for the transform set inTable 8.

While 16 transform coefficients are output when the RST is applied, onlym transform coefficients may be output when only an m×64 portion of a16×64 matrix is applied. For example, when only eight transformcoefficients are output by setting m=8 and multiplying only an 8×64matrix from the top, it is possible to reduce computational amount byhalf. To reduce computational amount in a worst case, an 8×64 matrix maybe applied to an 8×8 transform unit (TU).

An m×64 transform matrix applicable to an 8×8 region (m≤16, e.g., thetransform kernel matrices in Table 8) receives 64 pieces of data andgenerates m coefficients. That is, as shown in Equation 5, when the 64pieces of data form a 64×1 vector, an m×1 vector is generated bysequentially multiplying an m×64 matrix and a 64×1 vector. Here, the 64pieces of data forming the 8×8 region may be properly arranged to form a64×1 vector. For example, as shown in Table 10, the data may be arrangedin the order of indexes indicated at respective positions in the 8×8region.

TABLE 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 2425 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 4849 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

As shown in Table 10, the data is arranged in the row-first direction inthe 8×8 region for a secondary transform. This order refers to an orderin which two-dimensional data is one-dimensionally arranged for asecondary transform, specifically an RST or an LFNST, and may be appliedto a forward secondary transform performed in an encoding apparatus.Accordingly, in an inverse secondary transform performed by the inversetransformer of the encoding apparatus or the inverse transformer of thedecoding apparatus, transform coefficients generated as a result of thetransform, that is, primary transform coefficients, may betwo-dimensionally arranged as shown in Table 10.

When there are 67 intra prediction modes as shown in FIG. 5, alldirectional modes (mode 2 to mode 66) are symmetrically configured aboutmode 34. That is, mode (2 +n) is symmetric to mode (66−n) (0≤n≤31) aboutmode 34 in terms of prediction direction. Therefore, if a dataarrangement order for configuring a 64×1 input vector for mode (2 +n),that is, modes 2 to 33, corresponds to the row-first direction as shownin Table 10, a 64×1 input vector for mode (66−n) may be configured in anorder shown in Table 11.

TABLE 11 1 9 17 25 33 41 49 57 2 10 18 26 34 42 50 58 3 11 19 27 35 4351 59 4 12 20 28 36 44 52 60 5 13 21 29 37 45 53 61 6 14 22 30 38 46 5462 7 15 23 31 39 47 55 63 8 16 24 32 40 48 56 64

As shown in Table 11, the data is arranged in the column-first directionin the 8×8 region for a secondary transform. This order refers to anorder in which two-dimensional data is one-dimensionally arranged for asecondary transform, specifically an RST or an LFNST, and may be appliedto a forward secondary transform performed in an encoding apparatus.Accordingly, in an inverse secondary transform performed by the inversetransformer of the encoding apparatus or the inverse transformer of thedecoding apparatus, transform coefficients generated as a result of thetransform, that is, primary transform coefficients, may betwo-dimensionally arranged as shown in Table 11.

Table 11 shows that, for intra prediction mode (66−n), that is, formodes 35 to 66, a 64×1 input vector may be configured for according tothe column-first direction.

In summary, the same transform kernel matrix shown in Table 8 may beapplied while symmetrically arranging input data for mode (2+n)according to the row-first direction and input data for mode (66−n)(0≤n≤31) according to the column-first direction. A transform kernelmatrix to be applied in each mode is shown in Table 5 to Table 7. Here,either the arrangement order shown in Table 10 or the arrangement ordershown in Table 11 may be applied for the planar mode of intra predictionmode 0, the DC mode of intra prediction mode 1, and intra predictionmode 34. For example, for intra prediction mode 34, input data may bearranged according to the row-first direction as shown in Table 10.

According to another example, all of the illustrative transform kernelmatrices shown in Table 9 applicable to a 4×4 region are transformkernel matrices multiplied by 128 as a scaling value. In ag_aiNsst4×4[N1] [N2] [16] [64] array present in matrix arrays of Table9, N1 denotes the number of transform sets (N1 is 4 or 35, distinguishedby index 0, 1, . . . , and N1−1), N2 denotes the number (1 or 2)oftransform kernel matrices included in each transform set, and [16] [16]denotes a 16×16 transform.

As shown in Table 3 and Table 4, when a transform set includes onetransform kernel matrix, either a first transform kernel matrix or asecond transform kernel matrix may be used for the transform set inTable 9.

As in the 8×8 RST, only m transform coefficients may be output when onlyan m×16 portion of a 16×16 matrix is applied. For example, when onlyeight transform coefficients are output by setting m =8 and multiplyingonly an 8×16 matrix from the top, it is possible to reduce computationalamount by half. To reduce computational amount in a worst case, an 8×16matrix may be applied to a 4×4 transform unit (TU).

Basically, the transform kernel matrices applicable to a 4×4 region,presented in Table 9,may be applied to a 4×4 TU, a 4×M TU, and an M×4TU(M>4, the 4×M TU and the M×4 TU may be divided into 4×4 regions, towhich each designated transform kernel matrix may be applied, or thetransform kernel matrices may be applied only to a maximum top-left 4×8or 8×4 region) or may be applied only to a top-left 4×4 region. If thesecondary transform is configured to be applied only to the top-left 4×4region, the transform kernel matrices applicable to an 8×8 region, shownin Table 8, may be unnecessary.

An m×16 transform matrix applicable to a 4×4 region (m≤16, e.g., thetransform kernel matrices in Table 9) receives 16 pieces of data andgenerates m coefficients. That is, when the 16 pieces of data form a16×1 vector, an m×1 vector is generated by sequentially multiplying anm×16 matrix and a 16×1 vector. Here, the 16pieces of data forming the4×4 region may be properly arranged to form a 16×1 vector. For example,as shown in Table 12, the data may be arranged in the order of indexesindicated at respective positions in the 4×4 region.

TABLE 12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

As shown in Table 12, the data is arranged in the row-first direction inthe 4×4 region for a secondary transform. This order refers to an orderin which two-dimensional data is one-dimensionally arranged for asecondary transform, specifically an RST or an LFNST, and may be appliedto a forward secondary transform performed in an encoding apparatus.Accordingly, in an inverse secondary transform performed by the inversetransformer of the encoding apparatus or the inverse transformer of thedecoding apparatus, transform coefficients generated as a result of thetransform, that is, primary transform coefficients, may betwo-dimensionally arranged as shown in Table 12.

When there are 67 intra prediction modes as shown in FIG. 5, alldirectional modes (mode 2 to mode 66) are symmetrically configured aboutmode 34. That is, mode (2+n) is symmetric to mode (66−n) (0≤n≤31) aboutmode 34 in terms of prediction direction. Therefore, if a dataarrangement order for configuring a 16×1 input vector for mode (2 +n),that is, modes 2 to 33, corresponds to the row-first direction as shownin Table 12, a 64×1 input vector for mode (66−n) may be configured in anorder shown in Table 13.

TABLE 13 1 5 9 13 2 6 10 14 3 7 11 15 4 8 12 16

As shown in Table 13, the data is arranged in the column-first directionin the 4×4 region for a secondary transform. This order refers to anorder in which two-dimensional data is one-dimensionally arranged for asecondary transform, specifically an RST or an LFNST, and may be appliedto a forward secondary transform performed in an encoding apparatus.Accordingly, in an inverse secondary transform performed by the inversetransformer of the encoding apparatus or the inverse transformer of thedecoding apparatus, transform coefficients generated as a result of thetransform, that is, primary transform coefficients, may betwo-dimensionally arranged as shown in Table 13.

Table 13 shows that, for intra prediction mode (66−n), that is, formodes 35 to 66, a 16×1 input vector may be configured for according tothe column-first direction.

In summary, the same transform kernel matrices shown in Table 9may beapplied while symmetrically arranging input data for mode (2 +n)according to the row-first direction and input data for mode (66−n)(0≤n≤31) according to the column-first direction. A transform kernelmatrix to be applied in each mode is shown in Table 5 to Table 7. Here,either the arrangement order shown in Table 12 or the arrangement ordershown in Table 13 may be applied for the planar mode of intra predictionmode 0, the DC mode of intra prediction mode 1, and intra predictionmode 34. For example, for intra prediction mode 34, input data may bearranged according to the row-first direction as shown in Table 12.

On the other hand, according to another embodiment of this document, for64 pieces of data forming an 8×8 region, not the maximum 16×64 transformkernel matrix in Tables 8 and 9, but a maximum of 16×48 kernel matrixcan be applied by selecting only 48 pieces of data. Here, “maximum”means that the maximum value of m is 16 for an m×48 transform kernelmatrix capable of generating m coefficients.

A 16×48 transform kernel matrix according to the present embodiment maybe represented as shown in Table 14.

TABLE 14 const int g_aiNsst8x8[4][2][16][48] = {  { //0  {   {−117,28,18,2,4,1,2,1,32,−18,−2,0,−1,0,0,0,14,−1,−3,0,−1,0,0,0,2,0,0,0,0,0,0,0,3,0,−1,0,1,0,0,0,1,0,0,0,1,0,0,0},   {−29,−91,47,1,9,0,3,0,−54,26,−8,3,0,1,0,0,33,5,−9,−1,−2,0,−1,0,−3,3,0,0,0,0,0,0,7,2,−2,0,−1,1,0,0,2,1,−1,0,0,0,0,0},   {−10,62,−11,−8,−2,−2,−1,−1,−95,3,32,0,4,0,2,0,32,−30,−4,4,−1,1,0,0,6,2,−5,0,0,0,0,0,6,−3,0,0,2,0,−1,0,2,−1,0,0,1,0,0,0},   {−15,15,−10,−2,1,0,1,0,10,112,−20,−17,−4,−4,−1,−2,−20,−26,31,1,0,0,0,0,2,−16,−1,6,0,1,0,0,1,−4,0,0,0,−3,0,1,0,−1,0,0,0−2,0,0},   {32,39,92,−44,4,−10,1,−4,26,12,−15,13,5,2,−2,0,29,−16,−22,8,0,1,0,1,−20,6,4,−3,1,0,0,0,1,−4,−3,2,−4,1,0,0,1,−1,−2,1,−2,0,0,0},   {−10,1,50,−15,2,−3,1,−1,−28,−15,14,6,1,1,1,0,−99,−4,9,5,5,2,2,1,44,−10,−11,1,−2,0,−1,0,−5,4,−3,0,8,−1,−2,0,−2,1,−1,0,4,0,−1,0},   {1,−33,−11,−14,7,−2,2,0,29,−12,37,−7,−4,0,−1,0,6,−99,3,26,−1,5,0,2,14,30,−27,−2,1,−1,0,−1,−6,6,6,−3,1,3,−3,0,−1,1,1,0,0,1,−1,0},   {0,6,−6,21,−4,2,0,0,−20,−24,−104,30,5,5,1,2,−7,−46,10,−14,7,0,1,0,9,21,7,−6,−2,−1,0,−1,2,2,5,−2,0,3,4,−1,0,0,1,0,0,1,2,−1},   {−13,−13,−37,−101,29,−11,8,−3,−12,−15,−20,2,−11,5,−2,1,−12,10,26,12,−6,0,−1,0,−32,−2,11,3,3,−1,1,0,11,−5,−1,6,−4,2,1,0,3,−1,1,2,−1,0,0,0 },   {6,1,−14,−36,9,−3,2,0,10,9,−18,−1,−3,1,0,0,38,26,−13,−1,−5,−1,−1,0,102,3,−14,−1,−5,−1,−2,0,29,10,10,0,10,−4,−1,1,−7,1,2,1,2,−1,0,0 },   {−12,−2,−26,−12,−9,2,−1,1,−3,30,4,34,−4,0,−1,0,−30,3,−92,14,19,0,3,0,−11,34,21,−33,1,−2,0,−1,−9,−4,18,3,2,0,0,−2,−1,−1,3,0,0,0,0,−1 },   {0,−3,0,−4,−15,6,−3,1,−7,−15,−28,−86,19,−5,4,−1,−5,−17,−41,42,−6,2,−1,1,−1,−40,37,13,−4,2,−1,1,−10,13,−1,−4,4,−4,3,4,−2,2,−1,−1,1,−1,1,2 },   {−1,9,13,5,14,−2,2,−1,−8,3,−4,−62,4,1,1,0,−12,23,16,−11,−17,0,−1,0,−11,97,−3,−3,0,−6,0,−2,−21,−5,23,0,2,−2,−1,6,−3,−3,1,0,0,0,0,2 },   {6,2,−3,2,10,−1,2,0,8,3,−1,−20,0,1,0,0,−4,4,−16,0,−2,0,1,0,34,23,6,−7,−4,−2,−1,0,108,−5,−30,6,−27,10,7,−2,11,−3,−1,1,−4,1,0,1},   {6,9,−2,35,110,−22,11,−4,−2,0,−3,1,−18,12,−3,2,−5,−4,−22,8,−25,3,0,0,−3,−21,2,−3,9,−2,1,0,−7,1,3,−5,3,0,−1,0,0,1,0,−1,1,0,0,0},   {−1,7,−2,9,−11,5,−1,1,−7,2,−22,4,−13,0,−1,0,0,28,0,76,4,−6,0,−2,−13,5,−76,−4,33,−1,3,0,9,18,−3,−35,−4,−1,6,1,1,2,0,−3,−1,0,2,0},  },      {       {−108,48,9,1,1,1,0,0,44,−6,−9,−1,−1,0,−1,0,9,−9,−1,1,0,0,0,3,−1,1,0,0,0,0,0,1,−1,0,0,1,0,0,0,0,−1,0,0,0,0,0,0},       {55,66,−37,−5,−6,−1,−2,0,67,−30,−20,4,−2,0,−1,0,−31,−19,14,4,1,1,1,0,−6,3,5,−2,0,0,0,0,−7,−1,1,0,−1,1,1,0,−2,−1,1,0,0,0,0,0},       {2,86,−21,−13,−4,−2,−1,−1,−88,5,6,4,5,1,1,0,14,−5,0,3,0,0,0,0,10,−5,−2,0,−1,0,0,0,6,−5,0,1,2,−1,0,0,1,−1,0,0,1,0,0,0},       {−24,−21,−38,19,0,4,−1,2,−23,−89,31,20,2,3,1,1,−30,26,36,−8,−2,−2,0,−1,14,18,−7,−9,−1,−1,0,0,1,3,−2,−1,3,2,−2,−1,0,1,0,0,1,1,−   1,0 },       {9,20,98,−26,−3,−5,0,−2,−9,−26,15,−16,2,0,1,0,−61,−3,−2,3,7,1,1,0,12,16,−6,−1,0,−1,0,0,2,0,−8,1,3,1,−1,1,0,−1,−2,0,1,0,−1,0},       {−21,−7,−37,10,2,2,−1,1,−10,69,−5,−7,−2,−2,0,−1,−93,2,19,0,3,0,2,0,17,4,0,0,−1,0,0,0,5,−4,−2,0,4,−2,0,1,0,0,0,0,2,−1,0,0},       {−10,−25,4−17,8,−2,2,−1,−27,−17,−71,25,8,2,1,1,−4,−66,28,36,−5,3,0,1,−10,20,33,−13,−8,0,0,−1,3,6,−3,−7,−1,3,3,−1,1,0,−   1,0,0,1,1,−1 },       {2,5,10,64,−9,4,−3,1,−4,8,62,3,−17,1,−2,0,−3,−75,5,−14,1,4,0,1,−36,3,18,−4,4,0,1,0,14,−2,−8,−2,1,−3,0,2,2,−1,2,0,1,−1,0},       {−11,−15,−28,−97,6,−1,4,−1,7,3,57,−15,10,−2,0,−1,−1,−27,13,6,1,−1,0,0,−34,−6,0,3,4,1,2,0,−2,8,1,5,−2,0,−3,1,1,1,0,2,−1,0,−1,0},       {9,13,24,−6,7,−2,1,−1,16,39,20,47,−2,−2,−2,0,28,23,76,−5,−25,−3,−3,−1,6,36,−7,−39,−4,−1,0,−1,2,−4,−18,−3,−1,−1,−2,−2,1,−2,−   2,0,0,0,−1,−1 },       {−7,11,12,7,2,−1,0,−1,−14,−1,−24,11,2,0,0,0,−20,48,11,−13,−5,−2,0,−1,−105,−19,17,0,6,2,3,0,−14,8,8,2,1,2,−1,−2,3,0,−   1,0,0,0,0,0 },       {0,0,7,−6,23,−3,3,−1,5,1,18,96,13,−9,−1,−1,−21,−7,−42,14,−24,−3,0,0,11,−47,−7,3,−5,9,1,2,0,−1,19,−1,1,0,−1,−6,−1,1,2,0,1,0,0,−   2 },       {−2,−6,−1,−10,0,1,1,0,−7,−2,−28,20,−15,4,−3,1,−2,−32,−2,−66,3,7,1,2,−11,13,−70,5,43,−2,3,0,8,−14,−3,43,−1,2,7,−1,1,−2,1,3,−   1,1,1,0 },       {−1,6,−16,0,24,−3,1,−1,2,6,6,16,18,−7,1,−1,−3,11,−63,9,4,−5,2,−1,−22,94,−4,−6,−4,−4,1,−2,10,23,−19,−5,0,−6,−4,6,3,−2,1,1,0,−   1,0,0, },       {−5,−6,−3,−19,−104,18,−4,3,0,6,0,35,−41,20,−2,2,−2,10,−18,16,21,3,−2,0,−2,11,6,−10,6,−3,−1,0,−1,5,−1,−6,−1,−1,−1,−1,−   1,0,0,0,0,0,0,−1 },       {−1,−2,0,23,−9,0,−2,0,1,1,8,−1,29,1,1,0,3,−6,13,76,30,−11,−1,−2,−26,−8,−69,7,−9,−7,3,−1,−10,−34,−25,13,−1,0,11,5,1,−1,1,−   2,0,0,2,0 },      }     },     { //1      {       {110,−49,−3,−4,−1,−1,0,−1,−38,−1,10,0,2,0,1,0,−9,13,1,−2,0,0,0,0,−4,2,−3,0,0,0,0,0,−2,2,0,1,−1,1,0,0,−1,1,0,0,−1,0,0,0},       {−43,−19,17,−1,3,0,1,0,−98,46,14,−1,2,0,1,0,26,26,−15,−3,−2,−1,−1,0,11,−7,−9,2,0,0,0,0,9,−3,−1,2,3,−3,0,0,4,−1,0,0,2,−1,0,0},       {−19,17,−7,3,−2,1,−1,0,−32,−59,29,3,4,0,2,0,−72,43,34,−9,3,−2,1,−1,13,36,−18,10,0,−2,0,−1,3,0,−12,3,6,1,−3,2,1,−1,−2,0,3,1,−   1,1 },       {−35,−103,39,1,7,0,2,0,38,−13,25,−6,1,−1,0,0,−1,7,6,−7,1,−1,0,0,−13,14,2,−4,2,−1,0,0,−2,11,−6,−2,−2,4,−3,0,0,3,−2,0,−1,1,−1,0},       {9,5,−6,−1,−1,0,−1,0,42,4,21,−11,1,−3,1,−1,21,70,−32,−21,0,−4,−1,−1,34,−26,−57,11,4,2,0,1,−4,−32,5,24,1,−6,12,4,−3,−2,4,−2,0,−   1,0,0 },       {−5,−5,−28,9,−3,2,−1,1,−20,−78,22,16,1,3,0,1,80,−6,25,−5,−4,−1,−1,0,6,−24,7,−9,0,0,0,0,−7,3,13,−4,−3,5,1,−5,−2,3,1,−2,−1,2,−1,−   2 },       {14,17,27,−12,1,−3,1,−1,8,19,−13,4,−2,1,−1,0,48,−1,48,−15,−4,−2,−1,−1,1,60,−28,−42,5,−6,1,−2,11,−11,−51,11,−2,−10,−2,13,2,−   6,−4,4,−2,−3,2,2 },       {7,35,17,−4,−1,0,0,0,3,8,54,−17,1,−2,1,−1,10,14,−11,−34,4,−4,1,−1,−80,−7,−6,2,15,0,3,0,−16,46,1,3,2,7,−24,0,2,−2,−5,8,1,−1,−   2,2 },       {−13,−27,−101,24,−8,6,−3,2,11,43,6,28,−6,3,1,1,−3,14,21,−12,−7,−2,−1,−1,−23,10,−4,−12,3,0,1,0,2,9,−10,0,1,−5,−4,4,2,−   2,2,2,0,−2,1,0 },       {−11,−13,−3,−10,3,−1,1,0,−19,−19,−37,8,4,2,0,1,−12,−30,3,−9,5,0,1,0,−56,−9,−47,8,21,1,4,1,−11,−30,10,59,−2,8,41,8,2,5,6,−7,−   1,3,5,−2 },       {−4,−10,−24,−11,3,−2,0,−1,−6,−37,−45,−17,8,−2,2,−1,17,14,−58,14,15,0,2,0,−10,34,−7,28,4,−1,1,0,23,34,−31,4,10,−22,−30,22,4,−   15,9,20,2,−5,9,4 },       {−2,1,13,−17,3,−5,1,−2,3,0,−55,22,6,1,1,0,8,74,21,40,−14,0,−2,0,−36,−8,11,−13,23,1,−3,0,−36,6,16,−14,2,19,−4,−12,−1,0,−7,−   3,0,2,−2,−1 },       {3,1,5,−15,1,−2,1,−1,7,4,−7,29,−1,2,−1,1,8,3,12,−14,−9,1,−1,0,4,29,−15,31,10,4,1,1,61,22,55,14,13,3,−9,−65,1,−11,−21,−7,0,0,−   1,3 },       {−4,−8,−1,−50,6,−4,2,−2,−1,5,−22,20,6,1,0,0,−16,−15,18,−29,−11,2,−2,1,40,−45,−19,−22,31,2,4,1,−25,41,0,12,9,7,−42,12,−3,−   14,2,28,5,1,6,2 },       {5,−1,26,102,−13,12,−4,4,−4,−2,−40,−7,−23,3,−5,1,−1,5,8,−23,7,2,1,1,10,−11,−13,−3,12,−3,2,0,−9,23,4,9,14,9,−14,−4,0,−12,−   7,6,3,0,6,3 },       {−5,−6,−27,−22,−12,0,−3,0,−5,8,−20,−83,0,0,0,0,9,7,24,−20,41,3,6,1,15,20,12,11,17,−9,1,−2,−26,−1,18,−1,−12,32,3,−18,−5,10,−   25,−5,−2,1,−8,10 },      },     {      {80,−49,6,−4,1,−1,1,−1,−72,36,4,0,1,0,0,0,26,0,−12,2,−2,1,−1,0,−7,−9,6,1,0,0,0,0,3,5,−1,−2,−2,−2,−1,1,1,1,0,0,−1,−1,0,0},      {−72,−5,17,0,3,0,1,0,−23,58,−21,2,−3,1,−1,0,55,−46,−1,6,−2,1,−1,0,−22,7,17,−7,2,−1,1,0,9,5,−12,1,−3,−4,4,2,4,1,−2,−1,−1,−1,1,0},      {−50,19,−15,4,−1,1,−1,1,−58,−2,30,−3,4,−1,2,0,6,57,−34,0,−2,0,−1,0,34,−48,−2,14,−4,3,−1,1,−10,7,21,−10,6,1,−11,0,−1,−  1,4,2,3,0,−2,−1 },      {−33,−43,28,−7,4,−2,2,−1,−38,11,−8,4,1,1,0,0,−55,24,26,−5,2,−1,1,0,15,46,−40,−1,−1,0,−1,0,17,−38,1,17,−3,11,15,−11,3,−1,−  10,1,0,1,3,2 },      {10,66,−21,−3,−3,0,−1,0,−53,−41,−2,16,−1,4,−1,1,36,−5,41,−20,3,−3,1,−1,−30,26,−32,−3,7,−2,2,−1,15,−8,1,17,−1,−2,4,−8,2,0,−  1,3,0,0,0,−1 },      {18,14,13,−9,2,−2,1,−1,34,32,−31,12,−5,2,−2,1,40,4,−4,−9,−3,−2,−1,−1,27,−31,−43,19,−2,3,−1,1,7,−49,52,10,−11,22,7,−26,−1,−6,−  9,6,−2,2,4,−2 },      {21,66,−1,9,−4,2,−1,1,−21,41,−30,−10,0,−2,0,−1,−35,−17,−3,26,−6,5,−2,2,56,3,18,−25,−1,−2,−1,−1,−15,−13,−27,9,9,−6,20,5,−3,2,−  6,−9,3,−3,1,5 },      {1,−6,−24,17,−5,3,−2,1,24,10,39,−21,5,−4,2,−1,33,32,−30,4,−3,−1,−1,0,−4,13,−16,−10,0,−1,0,0,24,−26,−37,33,5,−32,55,−5,−7,22,−  14,−22,1,−9,−3,13 },      {9,33,−24,1,4,0,1,0,6,50,26,1,−10,0,−2,0,−27,1,−28,−21,16,−5,3,−2,23,36,−2,40,−17,4,−3,1,43,−13,4,−41,−19,−2,−24,17,11,−  4,8,4,−3,−3,−3,−3 },      {−7,−9,−32,14,−3,3,−1,1,−23,−28,0,−5,−1,0,0,0,−36,−59,−24,14,4,2,1,1,−23,−26,23,26,−3,5,0,2,10,−26,38,7,−12,11,42,−22,−  5,20,−14,−15,−1,−2,1,6 },      {6,30,69,18,5,−4,3,−1,−3,−11,−34,−16,9,−4,2,−1,−16,35,−35,30,−9,3,−2,1,−57,−13,6,4,−5,5,−1,1,28,10,4,7,0,−15,7,−10,−1,7,−  2,2,1,−3,0,0 },      {1,−8,24,−3,7,−2,2,−1,−6,−51,−6,−4,−5,0,−1,0,38,−1,0,25,6,2,1,1,47,20,35,1,−27,1,−5,0,37,−37,−9,−47,−28,5,0,18,8,6,0,−8,−4,−3,−  3,1 },      {4,10,4,17,−9,4,−2,1,5,14,32,−15,9,−3,2,−1,7,13,19,15,−8,1,−1,0,3,25,30,−18,1,−2,0,1,11,24,22,−11,−3,37,−13,−5,12,−  63,26,9,−15,11,8 },      {     −3,−9,−23,10,−10,3,−3,1,−5,−14,−16,−27,13,−5,2,−1,−1,−13,−30,11,−5,2,−1,0,−5,−8,−22,−16,10,0,1,0,0,−29,−27,6,−27,−10,−30,9,−  3,−10,−7,77,9,−13,45,−8 },      {2,11,22,2,9,−2,2,0,−6,−7,20,−32,−3,−4,0,−1,13,−5,−28,6,18,−4,3,−1,−26,27,−14,6,−20,0,−2,0,−76,−26,−4,−7,12,51,5,24,7,−17,−  16,−12,−5,4,2,13 },      {2,−3,8,14,−5,3,−1,1,−2,−11,5,−18,8,−3,2,−1,12,−23,−19,22,2,0,1,0,23,41,−7,35,−10,4,−1,1,5,7,23,5,69,−38,−8,−32,−15,−  31,24,11,2,1,18,11,−15 },     }    },     { //2      {       {−121,33,4,4,1,2,0,1,−1,−1,1,0,0,0,0,0,24,−5,−1,−1,0,0,0,0,5,−1,0,0,0,0,0,0,3,−1,0,0,2,−1,0,0,2,−1,0,0,1,0,0,0},       {0,−2,0,0,0,0,0,0,121,−23,−7,−3,−2,−1,−1,0,17,1,−2,0,0,0,0,0,−27,4,2,0,0,0,0,0,−12,2,1,0,−5,1,0,0,−1,0,0,0,−2,0,0,0},       {−20,19,−5,2,−1,1,0,0,16,3,−2,0,0,0,0,0,−120,14,8,1,3,1,1,0,−18,−2,3,0,1,0,0,0,17,−3,−1,0,6,−1,−1,0,2,0,0,0,2,0,0,0},       {32,108,−43,10,−9,3,−3,1,4,19,−7,1,−1,0,0,0,11,−30,9,−2,1,−1,0,0,0,−8,2,0,0,0,0,0,−7,−1,2,0,−3,−1,1,0,−2,−2,1,0,0,0,0,0},       {−3,0,−1,0,0,0,0,0,−29,11,−2,1,0,0,0,0,1,2,7,−1,0,0,0,0,0,−117,12,9,1,3,0,1,0,−32,−3,3,0,1,2,−2,−1,0,7,0,0,0,1,0,0,0},       {−4,−12,−3,1,−1,0,0,0,19,105,−31,7,−6,1−2,0,9,46,−6,0,0,0,0,0,8,−29,9,−3,1,0,0,0,−3,−19,3,0,−4,−6,1,0,0,0,0,0,0,−1,0,0},       {7,1,2,0,0,0,0,0,4,3,−2,0,0,0,0,0,22,−8,1,−1,0,0,0,0,−28,−9,4,01,0,0,0,117,−10,−8,0,3,2,1,−4,0,3,1,−1,0,−3,1,0,0},       {−8,−31,14,−4,3,−1,1,0,9,43,0,1,−1,0,0,0,−13,−105,17,−2,2,0,0,0,−8,−25,−3,0,0,0,0,0,−7,32,−5,1,−1,4,0,0,2,−1,0,0,1,0,−1,0},       {−15,−43,−100,23,−12,6,−4,2,−6,−17,−48,10,−5,2,−1,1,1,−5,19,−6,3,−1,1,0,2,7,15,−3,1,−1,0,0,4,10,5,−1,0,3,1,0,−2,1,2,0,−   1,1,1,0 },       {−3,1,2,0,0,0,0,0,−6,3,1,0,0,0,0,0,0,3,−2,0,0,0,0,0,−20,8,−2,0,0,0,0,0,30,13,−3,0,−116,6,10,0,−35,−5,4,0,−3,−1,0,0},       {−1,−6,−3,2,−1,0,0,0,−6,−35,9,0,2,0,0,0,1,−6,11,−2,2,0,1,0,−9,−100,17,−1,1,0,0,0−10,−63,1,2,−17,3,−4,0,−1,9,−1,0,3,4,−1,0},       {−5,−14,−48,2,−5,1,−2,0,10,24,99,−17,10,−4,3,−1,4,14,32,0,2,0,1,0,−4,0,−39,6,−4,1,−1,0,2,−3,−4,0,2,−2,−2,0,0,0,−1,0,0,−1,−1,0},       {−2,0,2,0,0,0,0,0,−2,0,1,0,0,0,0,0,−1,−1,1,−1,0,0,0,0,−1,−4,2,0,0,0,0,0,−8,−2,−1,1,30,4,−4,1,−102,4,8,−1,−69,−2,6,−1},       {−2,−10,−4,0,0,0,0,0,3,11,−1,−1,0,0,0,0,−6,−40,−15,6,−2,1,0,0,5,57,−6,2,0,0,0,0,1,−95,18,−6,−10,−34,−2,0,−4,17,−2,0,0,2,1,0},       {−2,−3,−25,−2,−3,0,−1,0,−1,−3,−1,−4,−2,2,0,1,−7,−8,−97,17,−9,3,−3,1,−8,−26,−61,−1,−3,−1,−1,−1,2,10,24,−7,5,9,19,−1,0,1,4,0,−   2,0,1,0 },       {4,−4,28,103,−42,24,−9,7,1,2,4,0,3,−1,0,0,−1,0,−9,−42,17,−9,3,−2,−1,1,−14,6,−4,2,−1,0,−1,−2,−4,4,0,3,1,−1,0,2,0,−2,2,0,0,0},      },     {      {87,−41,3,−4,1,−1,0,−1,−73,28,2,1,1,1,0,0,30,−5,−6,1,−1,0,0,0,−8,−3,3,0,0,0,0,0,3,2,−1,0,−2,−1,0,0,1,1,0,0,−1,0,0,0},      {−75,4,7,0,2,0,1,0,−41,36,−7,3,−1,1,0,0,72,−29,−2,0,−1,0,−1,0,−37,6,7,−2,1,0,0,0,12,3,−4,0,−3,−2,1,0,4,0,0,0,−1,0,0,0},      {26,−44,22,−6,4,−2,1,−1,77,24,−22,2,−4,0,−1,0,7,−38,10,0,1,0,0,0,−51,27,4,−3,2,−1,1,0,31,−5,−8,3,−14,0,5,−1,6,1,−3,0,−4,−  1,1,0 },      {−39,−68,37,−7,6,−2,2,0,−9,56,−21,1,−2,0,−1,0,−45,4,−3,6,−1,2,0,1,49,−13,3,−3,−1,0,0,0,−19,2,0,05,1,1,0,−2,0,−1,0,1,0,0,0},      {10,−20,2,0,1,0,0,0,50,−1,8,−5,1,−1,0,0,66,17,−24,4,−3,1,−1,0,13,−49,15,1,0,0,0,0,−53,34,6,−5,30,−7,−11,3,−11,−2,5,1,4,2,−1,−  1 },      {−21,−45,8,−2,3,−1,1,0,−7,−30,26,−8,3,−1,1,−1,−9,69,−33,5,−2,0,−1,0,−44,−31,10,7,−2,2,0,1,49,7,2,−6,−23,−3,−2,2,9,4,0,0,−2,−1,−  1,0 },      {−4,−2,−55,28,−8,5,−3,2,−2,37,43,−19,1,−2,1,−1,−47,−34,−27,5,4,−1,1,0,−39,−2,27,4,−2,1,0,0,−11,32,−8,−7,27,−12,−6,6,−13,0,4,−  3,3,−1,−2,1 },      {2,19,47,−23,6,−4,2,−1,−23,−22,−44,17,−2,2,−1,0,−33,3,22,−2,−4,1,−1,0,−58,−17,6,−6,7,−1,1,0,−23,40,−2,5,43,−11,−8,−1,−18,−  4,5,2,4,3,0,−1 },      {−19,−62,−9,9,0,0,0,0,−12,−56,27,−7,3,−1,1,0,7,−8,16,−6,4,−2,1,−1,−15,54,−23,2,−1,0,0,0,−42,−25,4,6,34,8,2,−2,−15,−1,0,−  1,3,2,0,1 },      {1,9,−5,0,−1,0,0,0,0,22,−1,2,0,1,0,0,−13,17,0,−2,0,−1,0,0,−46,−10,−10,4,−1,1,0,0,−80,−27,20,−4,−66,23,−2,−2,20,−3,−2,3,−  14,2,3,−1 },      {5,17,−9,0,−2,1,0,0,13,54,−2,7,−1,1,0,0,4,51,−3,−6,−1,−1,0,0,−20,6,−34,9,−2,2,−1,0,16,−52,28,1,59,15,−8,−5,−28,−7,2,2,10,3,0,−  1 },      {7,27,56,−2,10,−3,3,−1,−2,−6,8,−28,3,−4,1,−1,−1,−4,−68,35,−5,5,−2,1,0,35,43,−4,−6,1,−1,0,−14,−38,−12,−10,9,5,7,6,−9,7,−4,−3,4,−  4,0,3 },      {0,0,19,−4,3,−2,2,−1,−3,−13,10,−4,1,0,0,0,−6,−37,−18,−5,2,−2,1,−1,6,−6,−7,25,−6,4,−1,1,16,10,55,−24,15,46,−52,1,35,−43,10,12,−  23,13,5,−8 },      {3,0,−27,−80,40,−16,6,−4,4,3,31,61,−22,7,−1,1,−4,−7,−26,−6,−10,6,−4,1,3,8,14,−18,15,−5,2,−1,−2,−4,−1,13,0,2,−4,−3,3,−1,2,1,−  2,0,−2,−1 },      {1,2−8,6,−1,1,0,0,2,8,−5,−1,0,0,0,0,1,24,3,5,−1,1,0,0,−3,12,6,−10,1,−1,0,0,−9,−1,−25,10,45,−11,18,2,86,1,−13,−4,−65,−6,7,2},      {−4,−18,−57,8,−8,1,−3,0,−5,−20,−69,7,−6,2,−2,1,1,4,0,33,−7,5,−2,1,0,−9,53,−22,3,−1,0,0,4,−27,−2,−9,5,36,−13,5−7,−17,1,2,4,6,4,−  1 },     }    },     { //3      {       {−115,37,9,2,2,1,1,0,10,−29,8,0,1,0,1,0,23,−8,−8,1,−1,0,0,0,3,3,−2,−1,0,0,0,0,4,0,0,−1,1,1,0,0,2,0,0,0,0,0,0,0},       {15,51,−18,0,−3,0,−1,0,−95,7,34,−3,5,−1,2,0,23,−47,1,6,0,1,0,1,8,5,−12,0,−1,0,0,0,3,−3,1,−1,2,1,−2,0,1,−1,0,0,1,1,−1,0},       {29,−22,16,−6,3,−2,1,−1,−4,−80,12,15,0,3,0,1,45,7,−59,7,−2,1,−1,0,−15,41,−3,−16,2,−3,0,−1,1,0,7,−2,−3,6,1,−2,0,0,1,0,−1,2,0,−1},       {−36,−98,25,5,4,1,2,1,−59,11,−17,1,1,1,0,0,6,−13,7,−3,0,0,0,0,14,−4,−14,3,−1,0,0,0,2,8,−3,−5,2,0,0,0,0,3,0,−1,1,0,0,0},       {−6,18,3,−3,1,0,0,0,−50,−5,−38,12,0,2,0,1,3,67,−7,−40,3,−6,1,−3,−12,−13,65,−3,−10,0,−1,0,9,−20,−5,22,−2,0,0,−1,2,−3,−2,3,−   1,0,1,0 },       {4,15,52,−13,5,−3,2,−1,−17,−45,16,24,−2,4,−1,2,−87,−8,−14,7,8,1,2,0,23,−35,−6,−3,1,1,0,0,2,5,−17,0,3,−1,−1,−5,0,1,−4,0,1,0,0,−   2 },       {−20,−7,−43,4,0,1,−1,1,−7,35,0,12,−4,1,−1,0,−51,−2,−57,5,15,0,4,0,7,39,5,−55,1,−7,1,−3,1,−10,41,2,4,−3,−2,3,−1,−2,7,1,1,−1,−   1,0 },       {4,29,1,26,−5,4,−2,1,−17,−7,−73,6,6,2,1,1,−5,21,−3,5,−1,−3,0,−1,−11,2,−52,−3,27,−2,5,0,0,27,8,−58,2,−5,25,3,0,3,0,−5,0,−2,7,0},       {12,13,10,2,−1,3,−1,1,17,−2,−46,12,7,0,2,0,16,−45,−9,−53,6,1,1,0,70,16,8,−4,−37,1,−7,0,−12,29,3,21,4,0,5,−1,−3,4,1,4,2,0,1,0},       {5,20,90,−17,4,−3,2,−1,6,66,8,28,−7,3,−1,1,29,5,−19,12,9,−1,1,0,−10,14,−1,−13,7,0,1,0,0,−6,13,−4,0,−4,1,5,0,−1,−1,1,0,−1,0,0},       {−3,−4,−34,−12,2,−1,−1,0,5,25,11,43,−10,4,−2,1,23,20,−40,12,21,−3,4,−1,25,−28,−10,5,8,6,0,2,−4,21,−64,−8,−5,19,10,−48,3,−   1,10,−3,0,4,3,−6 },       {−1,−3,2,19,−2,4,−1,2,9,3,−35,22,11,1,2,0,−7,−65,−19,−22,11,4,2,1,−75,−18,3,−1,−10,2,0,1,2,−35,−27,4,1,8,−17,−19,3,0,3,−   6,0,2,−1,−2 },       {10,−4,−6,12,5,1,1,0,11,−9,−12,−2,−7,0,−1,0,33,−10,−4,18,−4,4,−1,28,−72,1,−49,15,2,2,1,56,−23,22,−1,4,−1,−15,26,6,4,−   10,0,0,2,−3,2 },       {4,6,14,53,−4,4,0,2,0,−1,−20,−13,3,2,−1,1,−3,1,−5,35,−16,−6,−1,−2,46,29,13,21,37,−5,4,−1,−10,−53,−18,8,9,12,−41,−25,−2,2,13,−   16,4,1,−5,1 },       {2,9,13,37,19,6,2,2,−9,−3,−9,−28,−20,−4,−2,−1,1,18,9,28,24,6,2,2,−20,−5,−25,−33,−36,9,−2,2,−13,42,1,57,−22,−2,−25,−   28,5,6,19,−12,−5,−3,−2,4 },       {3,−3,12,84,−12,8,−2,3,6,13,50,−1,45,1,7,0,−2,18,−22,−37,−13,14,0,3,1,−12,−3,2,−15,−8,1,−1,19,14,−4,−12,−4,5,17,8,2,−4,−4,4,−   2,2,1,0 },      },      {       {109,−26,−8,−3,−2,−1,−1,0,−50,28,2,1,0,0,0,0,−18,−8,6,0,1,0,1,0,6,−2,−3,0,0,0,0,0,−3,2,1,−1,0,0,0,0,−2,0,0,0,0,0,0,0},      {−39,31,−5,2,−1,1,0,0,−95,6,18,0,4,0,1,0,32,−49,5,1,1,0,0,0,27,−1,−14,2,−2,1,−1,0,3,5,−3,−2,4,1,−1,−1,2,0,0,0,2,0,0,0},      {29,−3,−2,−2,0,0,0,0,0,−41,9,0,2,0,1,0,86,4,−33,2,−6,1,−2,0,−32,58,1,−7,0,−2,0,−1,−14,−8,20,0,−2,−3,0,4,−1,−1,0,0,−1,1,0,0},      {18,96,−23,2,−5,1,−2,0,−10,6,10,−2,1,−1,1,0,−14,26,2,−4,1,−1,0,0,−43,−9,35,−2,4,−1,1,0,14,−40,1,10,2,1,−10,1,2,−4,−1,−1,0,0,−   1,0 },      {−29,−60,16,−2,3,−1,1,0,−52,9,−17,5,−2,1,−1,1,13,56,−2,−9,0,−2,0,−1,−34,−18,41,0,3,0,1,0,19,−36,−10,13,3,6,−14,−1,3,1,−1,−   3,1,1,−1,−1 },      {−23,−5,15,5,−2,1,−1,1,2,79,−13,−4,−2,−1,−1,0,−9,1,5,−1,1,0,0,0,−4,49,2,−14,1,−3,0,−1,−31,−14,56,−1,13,−37,−4,20,−2,2,−   10,0,2,−4,0,−1 },      {−7,−3,12,−3,3,−1,1,0,−31,−62,8,7,0,2,0,1,−75,9,−45,5,−1,1,−1,0,14,35,0,−23,2,−5,1,−2,1,−8,32,−1,7,−12,−4,10,0,2,−6,−1,2,0,0,−   2 },      {1,−26,5,0,1,0,1,0,24,−3,43,−6,4,−2,1,−1,−7,−64,9,14,0,3,0,1,−12,−4,5,3,−1,1,0,0,8,−59,−3,26,14,6,−58,6,−5,17,−7,−18,3,3,−1,−   5 },      {11,14,6,−3,1,−1,1,0,10,−7,−9,3,−2,1,−1,0,22,21,1,−21,2,−4,1,−2,92,1,53,0,−9,1,−2,0,−21,−11,1,40,−5,−4,−24,5,−4,5,−6,−5,0,0,0,−   3 },      {−10,−11,−47,3,−4,1,−1,0,5,28,11,−2,−1,0,0,0,−12,−2,−38,2,0,1,0,0,16,38,11,−16,−1,−3,0,−2,12,−9,−22,7,−8,60,4,−36,−6,−   15,54,7,3,−7,−8,14 },      {−8,−24,−99,11,−10,3,−4,1,−5,−36,19,−26,4,−5,1,−2,0,25,41,5,−3,1,0,0,10,−5,−7,12,2,1,0,0,−1,1,9,−3,−3,−14,−3,12,2,4,−13,−2,−   1,3,2,−4 },      {−5,1,−1,0,1,0,0,0,−10,−14,−6,8,0,1,0,0,−17,−2,7,−5,3,−1,0,0,−16,13,3,31,−1,6,0,2,−93,−15,−46,3,23,−19,0,−47,8,4,8,3,2,3,0,0},      {1,12,−20,21,−4,5,−2,2,−5,−2,−75,9,−1,2,−1,1,−1,−2,−16,−4,0,−1,0,0,−7,7,−31,0,3,0,0,0,4,11,−12,4,−12,14,−50,−1,−8,32,−4,−   54,2,0,30,−15 },      {2,−9,18,8,−3,3,−1,1,3,−25,−62,−6,0,−2,0,−1,−6,−61,14,−51,2,−6,0,−2,−19,0,40,−7,−17,0,−3,0,13,−4,11,9,17,0,24,5,1,−   12,4,28,0,0,−15,8 },      {4,9,39,18,0,2,0,1,−6,−16,−22,−37,5,−5,1,−2,−5,15,63,9,−16,0,−3,0,18,42,−18,27,15,1,3,1,12,−34,9,−24,4,28,−2,4,−11,−   4,30,2,5,−13,−4,18 },      {−7,−2,15,−6,1,−1,1,−1,−11,−3,22,−14,0,−2,1,−1,−18,−7,30,−9,−4,0,−1,0,−35,23,23,10,−17,1,−3,0,−19,53,6,48,−65,12,−12,11,−8,−   16,10,−21,−2,−12,6,2 },      }     }   },

When the RST is performed by applying an m×48 transform matrix (m≤16) toan 8×8 region, 64 pieces of data are inputted and m coefficients may begenerated. Table 14 shows an example of a transform kernel matrix when mis 16, and 48 pieces of data is inputted and 16 coefficients aregenerated. That is, assuming that 48 pieces of data form a 48×1 vector,a 16×1 vector may be generated by sequentially multiplying a 16×48matrix and a 48×1 vector. At this time, 48 pieces of data forming an 8×8region may be properly arranged to form a 48×1 vector, and the inputdata can be arranged in the following order.

TABLE 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 2425 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

When the RST is performed, as shown in Table 14, when a matrix operationis performed by applying a maximum 16×48 transform kernel matrix, 16modified transform coefficients are generate, the 16 modified transformcoefficients can be arranged in the top-left 4×4 region according to thescanning order, and the top- right 4×4 region and the bottom-left 4×4region can be filled with zeros. Table 16 shows an example of thearrangement order of 16 modified transform coefficients generatedthrough the matrix operation.

TABLE 16 1 3 6 10 2 5 9 13 4 8 12 15 7 11 14 16

As shown in Table 16, the modified transform coefficient generated whenthe maximum 16×48 transform kernel matrix is applied can be filled inthe top-left 4×4 region according to the scanning order. In this case,the number of each position in the top-left 4×4 region indicates thescanning order. Typically, the coefficient generated from an innerproduct operation of the topmost row of the 16×48 transform kernelmatrix and the 48×1 input column vector is the first in the scanningorder. In this case, the direction of going down to the bottom row andthe scanning order may match. For example, a coefficient generated fromthe inner product operation between a 48×1 input column vector and ann-th row from the top in the 16×48 transform kernel matrix becomes then-th in the scanning order.

In the case of the maximum 16×48 transform kernel matrix, the 4×4 regionin the top-right of Table 16 is the region to which the secondarytransformation is not applied, so the original input data (primarytransform coefficient) is preserved as it is, and the 4×4 region in thetop-right 4×4 region and the bottom-left 4×4 region will be filled withzeros.

In addition, according to another embodiment, a scanning order otherthan the scanning order shown in Table 16 may also be applied. Forexample, a row-first direction or a column first direction may beapplied as a scanning order.

In addition, even if the 16×64 transform kernel matrix shown in Table 8is applied, 16 transform coefficients are equally generated, so the 16transform coefficients can be arranged in the scanning order shown inTable 16 and in the case of applying the 16×64 transform kernel matrixsince the matrix operation is performed using all 64 input data insteadof 48, zeros are filled in all 4×4 regions except for the top-right 4×4region. Also in this case, the scanning order in the diagonal directionas shown in Table 16 may be applied, and other scanning order such asthe row first direction or the column first direction be applied.

On the other hand, when inverse RST or LFNST is performed as an inversetransformation process performed by the decoding apparatus, the inputcoefficient data to which the inverse RST is applied includes a 1Dvector according to the arrangement order of Table 16, and the modifiedcoefficient vector obtained by multiplying the 1D vector and thecorresponding inverse RST matrix from the left can be arranged in a 2Dblock according to the arrangement order in Table 15.

Then, the inverse transformer 322 of the decoding apparatus may applythe transform kernel matrix to transform coefficients arranged in onedimension according to the scanning order in Table 16. That is, 48modified transform coefficients can be derived through the matrixoperation between the one dimensional transform coefficients arrangedaccording to the scanning order in Table 16 and the transform kernelmatrix based on the transform kernel matrix in Table 14. That is, theone-dimensional transform coefficients can be derived into the 48modified transform coefficients through the matrix operation with amatrix in which the transform kernel matrix in Table 14 is transposed.

The 48 modified transform coefficients derived in this way can bearranged in two dimensions as shown in Table 15 for the inverse primarytransform.

In summary, in the transformation process, when RST or LFNST is appliedto the 8×8 region, the transform operation is performed between 48transform coefficients among the transform coefficients of the 8×8region in the top-left, the top-right and the bottom-left 4×4 regions ofthe 8×8 region excluding the bottom-right 4×4 region of the 8×8 regionand the 16×48 transform matrix kernel. For the matrix operation, 48transform coefficients are inputted in a one-dimensional array in theorder shown in Table 15. When such a matrix operation is performed, 16modified transform coefficients are derived, and the modified transformcoefficients may be arranged in the form shown in Table 16 in thetop-left region of the 8×8 region.

Conversely, in the inverse conversion process, when inverse RST or LFNSTis applied to the 8×8 region, 16 transform coefficients corresponding tothe top-left of the 8×8 region among the transform coefficients of the8×8 region are input in a one-dimensional array form according to thescanning order shown in Table 16, so that the transform operation isperformed between the 48×16 transform kernel matrix and the 16 transformcoefficients. That is, the matrix operation in this case can beexpressed as (48×16 matrix)*(16×1 transform coefficient vector) =(48×1modified transform coefficient vector). Here, since the n x 1 vector canbe interpreted in the same meaning as the n×1 matrix, it may beexpressed as an n x 1 column vector. Also, *means matrix multiplicationoperation. When such a matrix operation is performed, the 48 modifiedtransform coefficients can be derived, and the 48 modified transformcoefficients may be arranged in the top-left, the top-right, and thebottom- left 4×4 regions excluding the bottom-right 4×4 region of the8×8 region as shown in Table 15.

Meanwhile, according to an embodiment, as shown in Table 15, dataarrangement in an 8×8 region for the secondary transformation is inrow-first order. When there are 67 intra prediction modes as shown inFIG. 5, all directional modes (mode 2 to mode 66) are symmetricallyconfigured about mode 34. That is, mode (2+n) is symmetric to mode(66−n) (0≤n≤31) about mode 34 in terms of prediction direction.Therefore, if a data arrangement order for configuring a 48×1 inputvector for mode (2+n), that is, modes 2 to 33, corresponds to therow-first direction as shown in Table 15, a 48×1 input vector for mode(66−n) may be configured in an order shown in Table 17.

TABLE 17 1 9 17 25 33 37 41 45 2 10 18 26 34 38 42 46 3 11 19 27 35 3943 47 4 12 20 28 36 40 44 48 5 13 21 29 6 14 22 30 7 15 23 31 8 16 24 32

As shown in Table 11, the data is arranged in the column-first directionin the 8×8 region for a secondary transform. Table 17 shows that, forintra prediction mode (66−n), that is, for modes 35 to 66, a 48×1 inputvector may be configured for according to the column-first direction.

In summary, the same transform kernel matrix shown in Table 14 may beapplied while symmetrically arranging input data for mode (2+n)according to the row-first direction and input data for mode (66−n)(0≤n≤31) according to the column-first direction. A transform kernelmatrix to be applied in each mode is shown in Table 5 to Table 7.

Here, either the arrangement order shown in Table 15 or the arrangementorder shown in Table 17 may be applied for the planar mode of intraprediction mode 0, the DC mode of intra prediction mode 1, and the intraprediction mode 34. For example, for the planar mode of intra predictionmode 0, the DC mode of intra prediction mode 1, and the intra predictionmode 34, input data may be arranged according to the row-first directionas shown in Table 15 and the arrangement order shown in Table 16 can beapplied to the derived transform coefficients. Alternatively, for theplanar mode of intra prediction mode 0, the DC mode of intra predictionmode 1, and the intra prediction mode 34, input data may be arrangedaccording to the column-first direction as shown in Table 17 and thearrangement order shown in Table 16 can be applied to the derivedtransform coefficients.

As described above, when the 16×48 transform kernel matrix of Table 14is applied to the secondary transformation, the top-right 4×4 region andthe bottom-left 4×4 region of the 8×8 region are filled with zeros asshown in Table 16. When an m x 48 transform kernel matrix is applied tothe secondary transform (m≤16), not only the top-right 4×4 region andthe bottom-left 4×4 region, but also from the (m+1)th to 16th in thescanning order shown in Table 16 can be filled with zeros.

Therefore, if there is any non-zero transform coefficient from the(m+1)th to 16th position in the scanning order or in the top-right 4×4region or the bottom-left 4×4 region, it can be considered that the m×48secondary transform is (m≤16) is not applied. In this case, the indexfor the secondary transformation may not be signaled. The decodingapparatus first parses the transform coefficient and checks whether thecorresponding condition (that is, if a non-zero transform coefficientexists in the region where the transform coefficient should be 0) issatisfied and if it is satisfied the decoding apparatus may infer theindex for the secondary transformation to zero without parsing theindex. For example, in the case of m=16, it may be determined whether toapply the secondary transformation and whether to parse the index forthe secondary transformation by checking whether there is a non-zerocoefficient in the top-right 4×4 region or the bottom-left 4×4 region.

Meanwhile, Table 18 shows another example of transform kernel matricesthat can be applied to a 4×4 region.

TABLE 18 const int g_aiNsst4x4[4][2][16][16] = {  { //0   {    {108,−44,−15,1,−44,19,7,−1,−11,6,2,−1,0,−1,−1,0 },    {−40,−97,56,12,−11,29,−12,−3,18,18,−15,−3,−1,−3,2,1 },    {25,−31,−1,7,100,−16,−29,1,−54,21,14,−4,−7,2,4,0 },    {−32,−39,−92,51,−6,−16,36,−8,3,22,18,−15,4,1,−5,2 },    {8,−9,33,−8,−16,−102,36,23,−4,38,−27,−5,5,16,−8,6 },    {−25,5,16,−3,−38,14,11,−3,−97,7,26,1,55,−10,−19,3 },    {8,9,16,1,37,36,94,−38,−7,3,−47,11,−6,−13,−17,10 },    {2,34,−5,1,−7,24,−25,−3,8,99,−28,−29,6,−43,21,11 },    {−16,−27,−39,−109,6,10,16,24,3,19,10,24,−4,−7,−2,−3 },    {−9,−10,−34,4,−9,−5,−29,5,−33,−26,−96,33,14,4,39,−14 },    {−13,1,4,−9,−30,−17,−3,−64,−35,11,17,19,−86,6,36,14 },    {8,−7,−5,−15,7,−30,−28,−87,31,4,4,33,61,−5,−17,22 },    {−2,13,−6,−4,−2,28,−13,−14,−3,37,−15,−3,−2,107,−36,−24 },    {4,9,11,31,4,9,16,19,12,33,32,94,12,0,34,−45 },    {2,−2,8,−16,8,5,28,−17,6,−7,18,−45,40,36,97,−8 },    {0,−2,0,−10,−1,−7,−3,−35,−1,−7,−2,−32,−6,−33,−16,−112 },   },   {    {119,−30,−22,−3,−23,−2,3,2,−16,3,6,0,−3,2,1,0 },    {−27,−101,31,17,−47,2,22,3,19,30,−7,−9,5,3,−5,−1 },    {0,58,22,−15,−102,2,38,2,10,−13,5,4,14,−1,−9,0 },    {23,4,66,−11,22,89,−2,−26,13,−8,−38,−1,−9,−20,−2,8 },    {−19,−5,−89,2,−26,76,−11,−17,20,13,18,−4,1,−15,3,5 },    {−10,−1,−1,6,23,25,87,−7,−74,4,39,−5,0,−1,−20,−1 },    {−17,−28,12,−8,−32,14,−53,−6,−68,−67,17,29,2,6,25,4 },    {1,−24,−23,1,17,−7,52,9,50,−92,−15,27,−15,−10,−6,3 },    {−6,−17,−2,−111,7,−17,8,−42,9,18,16,25,−4,2,−1,11 },    {9,5,35,0,6,21,−9,34,44,−3,102,11,−7,13,11,−20 },    {4,−5,−5,−10,15,19,−2,6,6,−12,−13,6,95,69,−29,−24 },    {−6,−4,−9,−39,1,22,0,102,−19,19,−32,30,−16,−14,−8,−23 },    {4,−4,7,8,4,−13,−18,5,0,0,21,22,58,−88,−54,28 },    {−4,−7,0,−24,−7,0,−25,3,−3,−30,8,−76,−34,4,−80,−26 },    {0,6,0,30,−6,1,−13,−23,1,20,−2,80,−44,37,−68,1 },    {0,0,−1,5,−1,−7,1,−34,−2,3,−6,19,5,−38,11,−115 },   }  },      { //1      {        { −111,39,4,3,44,11,−12,−1,7,−16,−5,2,3,−1,4,2 },       { −47,−27,15,−1,−92,43,20,−2,20,39,−16,−5,10,−5,−13,2 },        {−35,−23,4,4,−17,−72,32,6,−59,18,50,−6,0,40,0,−13 },        {13,93,−27,−4,−48,13,−34,4,−52,11,1,10,3,16,−3,1 },        {−11,−27,1,2,−47,−4,−36,10,−2,−85,14,29,−20,−2,57,4 },        {0,−35,32,−2,26,60,−3,−17,−82,1,−30,0,−37,21,3,12 },        {−17,−46,−92,14,7,−10,−39,29,−17,27,−28,17,1,−15,−13,17 },        {4,−10,−23,4,16,58,−17,26,30,21,67,2,−13,59,13,−40 },        {5,−20,32,−5,8,−3,−46,−7,−4,2,−15,24,100,44,0,5 },        {−4,−1,38,−18,−7,−42,−63,−6,33,34,−23,15,−65,33,−20,2 },        {−2,−10,35,−19,5,8,−44,14,−25,25,58,17,7,−84,−16,−18 },        {5,13,18,34,11,−4,18,18,5,58,−3,42,−2,−10,85,38 },        {−5,−7,−34,−83,2,−1,−4,−73,4,20,15,−12,4,−3,44,12 },        {0,4,−2,−60,5,9,42,34,5,−14,9,80,−5,13,−38,37 },        {−1,2,7,−57,3,−7,9,68,−9,6,−49,−20,6,−4,36,−64 },        {−1,0,−12,23,1,−4,17,−53,−3,4,−21,72,−4,−8,−3,−83 },       },       {       { 88,−55,6,−3,−66,27,9,−2,11,11,−13,1,−2,−7,1,2 },        {−58,−20,27,−2,−27,75,−29,0,47,−42,−11,11,−9,−3,19,−4 },        {−51,23,−22,5,−63,3,37,−5,1,64,−35,−4,29,−31,−11,13 },        {−27,−76,49,−2,40,14,9,−17,−56,36,−25,6,14,3,−6,8 },        {19,−4,−36,22,52,7,36,−23,28,−17,−64,15,−5,−44,48,9 },        {29,50,13,−10,1,34,−59,1,−51,4,−16,30,52,−33,24,−5 },        {−12,−21,−74,43,−13,39,18,−5,−58,−35,27,−5,19,26,6,−5 },        {19,38,−10,−5,28,66,0,−5,−4,19,−30,−26,−40,28,−60,37 },        {−6,27,18,−5,−37,−18,12,−25,−44,−10,−38,37,−66,45,40,−7 },        {13,−28,−45,−39,0,−5,−39,69,−23,16,−12,−18,−50,−31,24,13 },        {−1,8,24,−51,−15,−9,44,10,−28,−70,−12,−39,24,−18,−4,51 },        {−8,−22,−17,33,−18,−45,−57,−27,0,−31,−30,29,−2,−13,−53,49 },        {1,12,32,51,−8,8,−2,−31,−22,4,46,−39,−49,−67,14,17 },        {4,5,24,60,−5,−14,−23,38,9,8,−34,−59,24,47,42,28 },        {−1,−5,−20,−34,4,4,−15,−46,18,31,42,10,10,27,49,78 },        {−3,−7,−22,−34,−5,−11,−36,−69,−1,−3,−25,−73,5,4,4,−49 },       }      },     { //2       {        { −112,47,−2,2,−34,13,2,0,15,−7,1,0,8,−3,−1,0},        { 29,−7,1,−1,−108,40,2,0,−45,13,4,−1,8,−5,1,0 },        {−36,−87,69,−10,−17,33,26,−2,7,14,−11,2,6,8,−7,0 },        {28,−5,2,−2,−29,13,−2,0,103,−36,−4,1,48,−16,−4,1 },        {−12,−24,15,−3,26,80,−61,9,15,54,−36,2,0,−4,6,−2 },        {18,53,69,−74,14,24,28,−30,−6,−7,−11,12,−5,−7,−6,8 },        {5,−1,2,0,−26,6,0,1,45,−9,−1,0,−113,28,8,−1 },       {−13,−32,18,−2,15,34,−27,7,−25,−80,47,−1,−16,−50,28,2 },        {−4,−13,−10,19,18,46,60,−48,16,33,60,−48,1,0,5,−2 },        {15,33,63,89,8,15,25,40,−4,−8,−15,−8,−2,−6,−9,−7 },        {−8,−24,−27,15,12,41,26,−29,−17,−50,−39,27,0,35,−67,26 },        {−2,−6,−24,13,−1,−8,37,−22,3,18,−51,22,−23,−95,17,17 },        {−3,−7,−16,−21,10,24,46,75,8,20,38,72,1,2,1,7 },        {2,6,10,−3,−5,−16,−31,12,7,24,41,−16,−16,−41,−89,49 },        {4,8,21,40,−4,−11,−28,−57,5,14,31,70,7,18,32,52 },        {0,1,4,11,−2,−4,−13,−34,3,7,20,47,−6,−19,−42,−101 },       },       {       { −99,39,−1,2,65,−20,−5,0,−15,−2,5,−1,0,3,−1,0 },        {58,42,−33,3,33,−63,23,−1,−55,32,3,−5,21,−2,−8,3 },        {−15,71,−44,5,−58,−29,25,3,62,−7,−4,−4,−19,4,0,1 },        {46,5,4,−6,71,−12,−15,5,52,−38,13,−2,−63,23,3,−3 },        {−14,−54,−29,29,25,−9,61,−29,27,44,−48,5,−27,−21,12,7 },        {−3,3,69,−42,−11,−50,−26,26,24,63,−19,−5,−18,−22,12,0 },        {17,16,−2,1,38,18,−12,0,62,1,−14,5,89,−42,8,−2 },        {15,54,−8,6,6,60,−26,−8,−30,17,−38,22,−43,−45,42,−7 },        {−6,−17,−55,−28,9,30,−8,58,4,34,41,−52,−16,−36,−20,16 },        {−2,−1,−9,−79,7,11,48,44,−13,−34,−55,6,12,23,20,−11 },        {7,29,14,−6,12,53,10,−11,14,59,−15,−3,5,71,−54,13 },|        {−5,−24,−53,15,−3,−15,−61,26,6,30,−16,23,13,56,44,−35 },        {4,8,21,52,−1,−1,−5,29,−7,−17,−44,−84,8,20,31,39 },        {−2,−11,−25,−4,−4,−21,−53,2,−5,−26,−64,19,−8,−19,−73,39 },        {−3,−5,−23,−57,−2,−4,−24,−75,1,3,9,−25,6,15,41,61 },        {1,1,7,18,1,2,16,47,2,5,24,67,3,9,25,88 },       }      },      { //3      {        { −114,37,3,2,−22,−23,14,0,21,−17,−5,2,5,2,−4,−1 },       { −19,−41,19,−2,85,−60,−11,7,17,31,−34,2,−11,19,2,−8 },        {36,−25,18,−2,−42,−53,35,5,46,−60,−25,19,8,21,−33,−1 },        {−27,−80,44,−3,−58,1,−29,19,−41,18,−12,−7,12,−17,7,−6 },        {−11,−21,37,−10,44,−4,47,−12,−37,−41,58,18,10,−46,−16,31 },        {15,47,10,−6,−16,−44,42,10,−80,25,−40,21,−23,−2,3,−14 },        {13,25,79,−39,−13,10,31,−4,49,45,12,−8,3,−1,43,7 },        {16,11,−26,13,−13,−74,−20,−1,5,−6,29,−47,26,−49,54,2 },        {−8,−34,−26,7,−26,−19,29,−37,1,22,46,−9,−81,37,14,20 },        {−6,−30,−42,−12,−3,5,57,−52,−2,37,−12,6,74,10,6,−15 },        {5,9,−6,42,−15,−18,−9,26,15,58,14,43,23,−10,−37,75 },        {−5,−23,−23,36,3,22,36,40,27,−4,−16,56,−25,−46,56,−24 },        {1,3,23,73,8,5,34,46,−12,2,35,−38,26,52,2,−31 },        {−3,−2,−21,−52,1,−10,−17,44,−19,−20,30,45,27,61,49,21 },        {−2,−7,−33,−56,−4,−6,21,63,15,31,32,−22,−10,−26,−52,−38 },        {−5,−12,−18,−12,8,22,38,36,−5,−15,−51,−63,−5,0,15,73 },       },       {       { −102,22,7,2,66,−25,−6,−1,−15,14,1,−1,2,−2,1,0 },        {12,93,−27,−6,−27,−64,36,6,1,3,5,−23,0,−2,6,5,−3 },        {−59,−24,17,1,−62,−2,−3,2,83,−12,−17,−2,−24,14,7,−2 },        {−33,23,−36,11,−21,50,35,−16,−23,−78,16,19,22,15,−30,−5 },        {0,−38,−81,30,27,5,51,−32,24,36,−16,12,−24,−8,9,1 },        {28,38,8,−9,62,32,−13,2,51,−32,15,5,−66,28,0,−1 },        {11,−35,21,−17,30,−18,31,18,−11,−36,−80,12,16,49,13,−32 },        {−13,23,22,−36,−12,64,39,25,−19,23,−36,9,−30,−58,33,−7 },        {−9,−20,−55,−83,3,−2,1,62,8,2,27,−28,7,15,−11,5 },        {−6,24,−38,23,−8,40,−49,0,−7,9,−25,−44,23,39,70,−3 },        {12,17,17,0,32,27,21,2,67,11,−6,−10,89,−22,−12,16 },        {2,−9,8,45,7,−8,27,35,−9,−31,−17,−87,−23,−22,−19,44 },        {−1,−9,28,−24,−1,−10,49,−30,−8,−7,40,1,4,33,65,67 },        {5,−12,−24,−17,13,−34,−32,−16,14,−67,−7,9,7,−74,49,1 },        {2,−6,11,45,3,−10,33,55,8,−5,59,4,7,−4,44,−66 },        {−1,1,−14,36,−1,2,−20,69,0,0,−15,72,3,4,5,65 },       }      }     };

The following embodiments may be proposed in order to reducecomputational amount in a worst case. In this document, a matrixincluding M rows and N columns is expressed as an M×N matrix, and theM×N matrix refers to a transform matrix applied in a forward transform,that is, when the encoding apparatus performs a transform (RST).Accordingly, in the inverse transform (inverse RST) performed by thedecoding apparatus, an N×M matrix obtained by transposing the M×N matrixmay be used. In addition, the following describes a case where an m×64transform kernel matrix (m≤16) is applied as a transformation matrix foran 8×8 region, but the same may be applied to a case where input vectoris 48×1 and the m×48 transform kernel matrix is (m≤16). That is, 16×64(or m×64) may be replaced with 16×48 (or m×48).

1) In a case of a block (e.g., a transform unit) having a width of W anda height of H where W >8 and H >8, a transform kernel matrix applicableto an 8×8 region is applied to a top-left 8×8 region of the block. In acase where W=8 and H=8, only an 8×64 portion of a 16×64 matrix may beapplied. That is, eight transform coefficients may be generated.Alternatively, only 8×48 parts of the 16×48 matrix can be applied. Thatis, 8 transform coefficients can be generated.

2) In a case of a block (e.g., a transform unit) having a width of W anda height of H where one of W and H is less than 8, that is, one of W andH is 4, a transform kernel matrix applicable to a 4×4 region is appliedto a top-left region of the block. In a case where W=4 and H=4, only an8×16 portion of a 16×16 matrix may be applied, in which case eighttransform coefficients are generated.

If (W, H)=(4, 8) or (8, 4), a secondary transform is applied only to thetop-left 4×4 region. If W or H is greater than 8, that is, if one of Wand H is equal to or greater than 16 and the other is 4, the secondarytransform is applied only to two top-left 4×4 blocks. That is, onlyatop-left 4×8 or 8×4 region may be divided into two 4×4 blocks, and adesignated transform kernel matrix may be applied thereto.

3) In a case of a block (e.g., a transform unit) having a width of W anda height of H where both W and H are 4, a secondary transform may not beapplied.

4) In a case of a block (e.g., a transform unit) having a width of W anda height of H, the number of coefficients generated by applying asecondary transform may be maintained to be 1/4 or less of the region ofthe transform unit (i.e., the total number of pixels included in thetransform unit=W×H). For example, when both W and H are 4, a top 4×16matrix of a 16×16 matrix may be applied so that four transformcoefficients are generated.

Assuming that a secondary transform is applied only to a top-left 8×8region of the entire transform unit (TU), eight or less coefficientsneed to be generated for a 4×8 transform unit or a 8×4 transform unit,and thus a top 8×16 matrix of a 16×16 matrix may be applied to a topleft 4×4 region. Up to a 16×64 matrix (or 16×48 matrix) may be appliedto an 8×8 transform unit (up to 16 coefficients can be generated). In a4×N or N×4 (N >16) transform unit, a 16×16 matrix may be applied to atop-left 4×4 block, or a top 8×16 matrix of the 16×16 matrix may beapplied to two top-left 4×4 blocks. Similarly, in a 4×8 transform unitor 8×4 transform unit, eight transform coefficients may be generated byapplying a top 4×16 matrix of the 16×16 matrix to two top-left 4×4blocks.

5) The maximum size of a secondary transform applied to a 4×4 region maybe limited to 8×16. In this case, the amount of a memory required tostore transform kernel matrices applied to the 4×4 region can be reducedby half compared to that in a 16×16 matrix.

For example, in all transform kernel matrices shown in Table 9 or Table18, the maximum size may be limited to 8×16 by extracting only a top8×16 matrix of each 16×16 matrix, and an actual image coding system maybe implemented to store only 8×16 matrices of the transform kernelmatrices.

If the maximum applicable transform size is 8×16 and the maximum numberof multiplications required to generate one coefficient is limited to 8,an up to 8×16 matrix may be applied to a 4×4 block, and an up to 8×16matrix may be applied to each of up to two top-left two 4×4 blocksincluded in a 4×N block or an N×4 block (N≥8, N=2n, n≥3). For example,an 8×16 matrix may be applied to one top-left 4×4 block in a 4 x N blockor an N×4 block (N≥8, N=2n, n≥3).

According to an embodiment, when coding an index specifying a secondarytransform to be applied to a luma component, specifically, when onetransform set includes two transform kernel matrices, it is necessary tospecify whether to apply the secondary transform and which transformkernel matrix to apply in the secondary transform. For example, when nosecondary transform is applied, a transform index may be coded as 0, andwhen the secondary transform is applied, transform indexes for twotransform sets may be coded as 1 and 2, respectively.

In this case, when coding the transform index, truncated unary codingmay be used. For example, binary codes of 0, 10, and 11 may berespectively allocated to transform indexes 0, 1, and 2, thereby codingthe transform indexes.

In addition, when coding the transform index by truncated unary coding,different CABAC context may be assigned to each bin. When coding thetransform indexes 0, 10, and 11 in the above example, two CABAC contextsmay be used.

When coding a transform index specifying a secondary transform to beapplied to a chroma component, specifically, when one transform setincludes two transform kernel matrices, it is necessary to specifywhether to apply the secondary transform and which transform kernelmatrix to apply in the secondary transform similarly to when coding thetransform index of the secondary transform for the luma component. Forexample, when no secondary transform is applied, a transform index maybe coded as 0, and when the secondary transform is applied, transformindexes for two transform sets may be coded as 1 and 2, respectively.

In this case, when coding the transform index, truncated unary codingmay be used. For example, binary codes of 0, 10, and 11 may berespectively allocated to transform indexes 0, 1, and 2, thereby codingthe transform indexes.

In addition, when coding the transform index by truncated unary coding,different CABAC context may be assigned to each bin. When coding thetransform indexes 0, 10, and 11 in the above example, two CABAC contextsmay be used.

According to an embodiment, a different CABAC context set may beallocated according to a chroma intra prediction mode. For example, whenchroma intra prediction modes are divided into non-directional modes,such as a planar mode or a DC mode, and other directional modes (i.e.,divided into two groups), a corresponding CABAC context set (includingtwo contexts) may be allocated for each group when coding 0, 10, and 11in the above example.

When the chroma intra prediction modes are divided into a plurality ofgroups and a corresponding CABAC context set is allocated, it isnecessary to find out a chroma intra prediction mode value before codingthe transform index of a secondary transform. However, in a chromadirect mode (DM), since a luma intra prediction mode value is used as itis, it is also necessary to find out an intra prediction mode value fora luma component. Therefore, when coding information on a chromacomponent, data dependency on luma component information may occur.Thus, in the chroma DM, when coding the transform index of the secondarytransform without having information on the intra prediction mode, thedata dependency can be removed by mapping to a specific group. Forexample, if the chroma intra prediction mode is the chroma DM, thetransform index may be coded using a corresponding CABAC context setassuming the planner mode or the DC mode, or a corresponding CABACcontext set may be applied assuming that other directional modes.

Hereinafter, when the secondary transform such as RST or LFNST isconsidered, in forward image processing that generates transformcoefficients from a residual signal generated through prediction,transform coefficients are generated by applying the forward secondarytransform to the result value of the forward primary transform or theresult vector of the forward primary transform as input. In addition, ininverse image processing that generates a pixel-domain residual signalfrom a transform coefficient reconstructed through a dequantization, anoutput value is derived by applying the inverse secondary transform tothe transform coefficient or transform coefficient vector as an input,and the final residual signal is derived by applying the inverse primarytransform to this output again as an input.

The output of the forward primary transform and the input of the inverseprimary transform can be defined to have a certain range of values foreach element. Here, the output and the input may be a two-dimensionalblock, and the element may be each value constituting thetwo-dimensional block.

For example, if each element of the output or input has to be stored ina 16-bit signed integer variable expressed in 2's complement form, theallowable range is limited to −2¹⁵ to 2¹⁵−1. The output of the forwardsecondary transform, more specifically, each element constituting theoutput vector does not deviate from the output range of the forwardprimary transform, or the output of the inverse secondary transform,more specifically, each element constituting the output vector may berestricted so as not to deviate from the input range of the inverseprimary transform. A more specific example is as follow.

(1) Limit the output range of the forward secondary transformation

When an M×1 vector is output by applying the forward secondarytransform, for example, when RST for an 8×8 block is applied, a 16×1vector may be output by taking a 64×1 vector or a 48×1 vector as aninput, multiplying it by a 16×64 matrix or a 16×48 matrix, and thenapplying appropriate scaling and/or rounding. In this case, it ispossible to restrict each M element constituting the M×1 vector to havea minFwdSecTr value as a minimum value and a maxFwdSecTr value as amaximum value.

The following Equation represents the Clip3 function and can be used tolimit the output range of each element.

$\begin{matrix}{{{Clip}\; 3\left( {x,y,z} \right)} = \left\{ \begin{matrix}x & ; & {z < x} \\y & ; & {z > y} \\z & ; & {otherwise}\end{matrix} \right.} & \left\lbrack {{Equation}\mspace{14mu} 9} \right\rbrack\end{matrix}$

The Clip3(x, y, z) function shown in Equation 9 is a function thatconverges a scalar input value z, where x and y indicate the maximum andminimum values allowed, respectively. Accordingly, if each of the Nelements is denoted by z, a result value obtained by applying Clip3(maxFwdSecTr, minFwdSecTr, z) to each z can be output. According to theimage processing aspect, the result of applying Clip3(maxFwdSecTr,minFwdSecTr, z) can also be considered as the final result of theforward secondary transformation. Meanwhile, the scaling and roundingmay be performed by the following equation for each element.

y=(x+(1<<(S−1)))>>S   [Equation 10]

In Equation 10, x denotes an input value, y denotes an output value, andS can be set to 7 if the secondary transform matrix is scaled by 128.

Two variables maxFwdSecTr and minFwdSecTr shown in the forwardClip3(maxFwdSecTr, minFwdSecTr, z) function may be set as in Equation11.

maxFwdSecTr=2¹⁵−1

minFwdSecTr=−2¹⁵   [Equation 11]

When the variable is set as in Equation 11, each element constitutingthe output of the forward secondary transformation may be stored in a16-bit signed integer variable expressed in the form of 2′s complement.

When the element is to be stored in an N-bit signed integer variable,Equation 11 may be expressed as Equation 12.

maxFwdSecTr=2^(N−1)−1

minFwdSecTr=−2^(N−1)   [Equation 12]

(2) Limit the output range of the inverse secondary transformation

When an M×1 vector is output by applying inverse secondary transform,for example, when applying RST to an 8×8 block, a 16×1 vector may beoutput by taking a 64×1 vector or a 48×1 vector as an input, multiplyingit by a 16×64 matrix or a 16×48 matrix, and then applying appropriatescaling and/or rounding. In this case, it is possible to limit each Melement constituting the M×1 vector to have a minInvSecTr value as aminimum value and a maxInvSecTr value as a maximum value.

That is, if the Clip3 function of Equation 9 is applied, when each ofthe M elements is denoted by z, a result value obtained by applyingClip3(maxInvSecTr, minInvSecTr, z) to each z may be output. According toan image processing aspect, the result of applying Clip3( maxInvSecTr,minInvSecTr, z) may be regarded as the final result of the backwardsecondary transformation. Meanwhile, Equation 10 may be applied to thescaling and rounding for each element, and as described above, if thesecondary transformation matrix is scaled by 128, S may be set to 7.

Two variables maxInvSecTr and minInvSecTr shown in the reverseClip3(maxInvSecTr, minInvSecTr, z) function may be set as in Equation13.

maxInvSecTr=2¹⁵−1

minInvSecTr=−2¹⁵   [Equation 13]

When the variable is set as in Equation 13, each element constitutingthe output of the forward secondary transformation may be stored in a16-bit signed integer variable expressed in the form of 2's complement.

When the element is to be stored in an N-bit signed integer variable,Equation 13 may be expressed as Equation 14.

maxInvSecTr=2^(N−1)−1

minInvSecTr=−2^(N−1)   [Equation 14]

Both (1) Limit the output range of the forward secondary transformationand (2) Limit the output range of the inverse secondary transformationmay be applied in the transform process, or only one of them may beapplied. That is, the output range limitation of the forward secondarytransform may be applied only in the encoding process of an image, orthe output range limit of the inverse secondary transform may be appliedonly in the decoding process.

Hereinafter, the results of various tests employing an RST areillustrated.

According to an embodiment, Test 1 to Test 4 in which signaling ofMTS-related information (whether to apply an MTS and information on anMTS kernel) is not considered are conducted. The configurations of Test1 to Test 4 are as follows, and details of (A) to (D) for theconfigurations of the tests are present in Table 19.

1) Test 1: (A)

2) Test 2: (A)+(B)

3) Test 3: (A)+(B)+(C)

4) Test 4: (A)+(B)+(D)

TABLE 19 Feature Details (A) Reduce number of transform sets from 35 to4, and apply two transforms per transform set (B) Limited RST byapplying up to 8 multiplication operations per sample (C) Not apply RSTin 4 × 4 TU (D) Apply 16 × 48 transform matrix instead of 16 × 64transform matrix

The results of the tests are summarized as follows.

According to the result of Test 1, compared to a conventional case(e.g., VTM anchor) in which the foregoing conditions are not applied,encoding time is 128% (AI), 108% (RA), and 104% (LD), and a reduction inBD rate is −1.59% (AI), −0.88% (RA), and −0.24% (LD).

According to the result of Test 2, compared to the VTM anchor, encodingtime is 128% (AI), 108% (RA), and 104% (LD), and a reduction in BD rateis −1.40% (AI), −0.76% (RA), and −0.21% (LD).

According to the result of Test 3, compared to the VTM anchor, encodingtime is 125% (AI), 107% (RA), and 103% (LD), and a reduction in BD rateis −1.25% (AI), −0.69% (RA), and −0.18% (LD).

According to the result of Test 4, compared to the VTM anchor, encodingtime is 129% (AI), 107% (RA), and 104% (LD), and a reduction in BD rateis −1.34% (AI), −0.71% (RA), and −0.18% (LD).

The memory usages of Test 1, Test 2, and Test 3 are measured to be 10KB, and the memory usage of Test 4 is measured to be 8 KB.

In these tests, four features are proposed for a secondary transform.

1. Application of RST

For the secondary transform, four transform sets are applied instead of35 transform sets, and memory mapping for the four transform sets isshown in Table 20. 16×64 (16×48 in Test 4) and 16×16 transform matricesare applied to 8×8 and 4×4 blocks, respectively. For convenience, the16×64 (or 16×48) transform matrix may be referred to as an RST 8×8 andthe 16×16 transform matrix may be referred to as an RST 4×4.

TABLE 20 Intra mode 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 NSST Set0 0 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 Intra mode 18 19 20 21 22 23 24 2526 27 28 29 30 31 32 33 NSST Set 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 Intramode 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 NSST Set 3 3 3 33 3 3 3 3 3 3 2 2 2 2 2 2 Intra mode 51 52 53 54 55 56 57 58 59 60 61 6263 64 65 66 NSST Set 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1

As described above, an RST 8×8 having a reduced factor of 4, that is,reduced to a 1/4 size, may be applied to an 8×8 region. Accordingly, a16×64 matrix may be applied instead of a typical 8×8 non-separabletransform matrix having a size of 64×64. That is, in a decodingapparatus, a 64×16 inverse RST matrix is used to generate core transformcoefficients, that is, primary transform coefficients, in the 8×8region. A forward RST 8×8 uses a 16×64 or 8×64 matrix for an 8×8 blockand thus generates non-zero coefficients only in a top-left 4×4 regionin the 8×8 region. That is, when the RST is applied, the 8×8 regionexcluding the top-left 4×4 region may be filled with 0s. In an RST 4×4,a 16×16 or 8×16 matrix multiplication is performed for a 4×4 block.

An inverse RST may be conditionally performed only when a predeterminedcondition is satisfied. For example, the inverse RST may be performedwhen a block has a size equal to or greater than a preset thresholdvalue (W≥4 && H≥4) and when a transform skip mode flag indicating atransform skip mode is 0.

When both the width W and the height H of a transform coefficient blockare greater than 4, an inverse RST 8×8 is applied to a top-left 8×8region of the transform coefficient block. Otherwise, an RST 4×4 may beapplied to a top-left min(8, W)×min(8, H) region of the transformcoefficient block.

An RST index, that is, a transform index, of 0 may indicate that an RSTis not applied, and otherwise, a transform kernel may be selectedthrough the transform index.

Further, an RST may be applied to intra coding units existing in anintra slice and an inter slice and coding units for both luma andchroma. When the partition type of a coding unit is a dual tree,respective RST indexes (transform indexes) for a luma block and a chromablock may be signaled. For an inter slice to which the dual tree is notapplied, one RST index may be signaled and applied to both a luma blockand a chroma block.

When an RST is applied, a transform matrix for the RST is selected fromthe four transform sets as shown in Table 20, and each transform setincludes two transforms. Transform sets determined according to an intraprediction mode are differently expressed as shown in Table 21. In thiscase, when the intra prediction mode is any one of three CCLM modes(INTRA_LT_CCLM, INTRA_T_CCLM, and INTRA_L_CCLM), transform set 0 isselected, and otherwise a transform set may be mapped as shown in Table21.

TABLE 21 Tr. set IntraPredMode index IntraPredMode < 0 1 0 <=IntraPredMode <= 1 0 2 <= IntraPredMode <= 12 1 13 <= IntraPredMode <=23 2 24 <= IntraPredMode <= 44 3 45 <= IntraPredMode <= 55 2 56 <=IntraPredMode 1

According to Table 21, intra prediction modes ranging from −14 to −1 and67 to 80 are transform modes for wide-angle intra prediction.

2. Reduced operation

The number of multiplications for eight samples in a worst case isproposed to be 8 or less.

When an RST 8×8 and an RST 4×4 are used, the worst case of amultiplication occurs when all transform units includes 4×4 transformunits or 8×8 transform units. Thus, top 8×64 and 8×16 portions of amatrix, that is, eight transform basis vectors on the top of eachtransform matrix, are applied to a 4×4 transform unit or an 8×8transform unit.

When a block is greater than an 8×8 transform unit, the worst case doesnot occur, and thus an RST 8×8 (e.g., a 16×64 matrix) is applied to atop-left 8×8 region. For an 8×4 transform unit or a 4×8 transform unit(e.g., a 16×16 matrix), an RST 4×4 may be applied only to a top-left 4×4region except for other 4×4 regions in order to prevent the worst casefrom occurring. For an 4×N or N×4 transform unit (N≥16), an RST 4×4 isapplied to each of two adjacent 4×4 regions.

3. RST matrix in reduced dimension

In Test 4, a 16×48 transform matrix is applied instead of a 16×64transform matrix in the same transform set configuration. 48 pieces ofinput data are extracted from three 4×4 blocks except for a bottom-right4×4 block in a topmost 8×8 region of a block to be transformed. Due to areduction in dimension, memory usage required to store all RST transformmatrices is reduced from 10 KB to 8 KB with a reasonable level ofperformance deterioration.

FIG. 9 illustrates a forward RST 8×8 using a 16×48 transform matrixaccording to an embodiment of the present disclosure.

As shown, when a two-dimensional forward primary transform is performedon a residual block having both a width and a height greater than 8, ablock of M×N primary transform coefficients is generated. A 16×48secondary transform matrix may be applied to top-left, top-right, andbottom-left 4×4 blocks of a top-left 8×8 region of the block of theprimary transform coefficients excluding a bottom-right 4×4 block. Thatis, a 16×1 secondary transform coefficients are generated through thematrix multiplication of a 16×48 transform kernel and the 48×1 primarytransform coefficients. The generated 16 modified transform coefficientsmay be arranged in the top-left 4×4 region of the 8×8 region, and thetop-right 4×4 region and the bottom-left 4×4 region of the 8×8 regionmay be filled with 0s. The remaining region, to which an RST is notapplied in a secondary transform, is filled with primary transformcoefficients as they are.

The result of each test is shown in the following tables.

Table 22 to Table 26 show the result of Test 1 in which four transformsets and two transforms for each transform set are applied. Table 22 toTable 24 show the result of the test in which an inter MTS (applicationof an MTS to an inter coding unit) is off, and Table 25 and Table 26show the result of the test in which the inter MTS is on.

TABLE 22 All Intra Main10 Over VTM-3.0 Y U V EncT DecT Class A1 −1.95%−3.13% −2.90% 127% 94% Class A2 −0.93% −2.39% −1.97% 131% 97% Class B−1.20% −2.74% −3.58% 128% 98% Class C −1.90% −2.37% −3.36% 128% 100% Class E −2.14% −3.25% −3.99% 127% 97% Overall −1.59% −2.75% −3.22% 128%98% Class D −1.64% −2.43% −2.97% 128% 101%  Class F −1.85% −2.01% −2.36%131% 101% 

TABLE 23 Random access Main10 Over VTM-3.0 Y U V EncT DecT Class A1−1.25% −2.05% −2.16% 110%  99% Class A2 −0.71% −1.50% −1.28% 107% 100%Class B −0.72% −2.24% −3.31% 107% 101% Class C −0.91% −1.57% −1.80% 108%101% Class E Overall −0.88% −1.88% −2.27% 108% 100% Class D −0.67%−2.06% −2.36% 107% 100% Class F −0.98% −1.32% −1.67% 109% 101%

TABLE 24 Low delay B Main10 Over VTM-3.0 Y U V EncT DecT Class A1 ClassA2 Class B −0.22% −1.17% −1.45% 105%  99% Class C −0.30% −0.67% −0.31%106% 102% Class E −0.21% −0.44% −1.91% 102% 102% Overall −0.24% −0.83%−1.19% 104% 101% Class D −0.15% −0.76% −0.11% 107% 103% Class F −0.34%−0.24% −0.60% 106% 102%

TABLE 25 Random access Main10 Over VTM-3.0 Y U V EncT DecT Class A1−1.28% −2.06% −2.26% 108% 100% Class A2 −0.73% −1.65% −1.37% 105%  99%Class B −0.77% −2.40% −3.33% 106% 101% Class C −0.96% −1.48% −1.90% 106%101% Class E Overall −0.91% −1.94% −2.34% 106% 100% Class D −0.71%−2.03% −2.02% 106% 101% Class F −0.97% −1.37% −1.66% 107% 101%

TABLE 26 Low delay B Main10 Over VTM-3.0 Y U V EncT DecT Class A1 ClassA2 Class B −0.28% −0.97% −1.44% 103%  99% Class C −0.35% −0.87% −1.03%104% 101% Class E −0.21% −0.57% −1.72% 102% 101% Overall −0.29% −0.84%−1.37% 103% 100% Class D −0.14% −0.95% −0.67% 104% 101% Class F −0.49%−0.52% −0.23% 104% 101%

Table 27 to Table 31 show the result of Test 2 in which features (A) and(B) are combined. Table 27 to Table 29 show the result of the test inwhich an inter MTS is off, and Table 30 and Table 31 show the result ofthe test in which the inter MTS is on.

TABLE 27 All Intra Main10 Over VTM-3.0 Y U V EncT DecT Class A1 −1.90%−3.09% −2.94% 127% 93% Class A2 −0.85% −2.23% −1.76% 130% 95% Class B−1.06% −2.60% −3.52% 128% 97% Class C −1.43% −2.20% −3.06% 128% 99%Class E −1.99% −3.02% −3.90% 127% 98% Overall −1.40% −2.60% −3.09% 128%97% Class D −1.01% −2.26% −2.73% 127% 101%  Class F −1.39% −1.94% −2.27%129% 100% 

TABLE 28 Random access Main10 Over VTM-3.0 Y U V EncT DecT Class A1−1.23% −1.89% −2.19% 110%  99% Class A2 −0.66% −1.51% −1.30% 106% 100%Class B −0.63% −2.37% −3.29% 107% 100% Class C −0.63% −1.36% −1.73% 108%101% Class E Overall −0.76% −1.83% −2.25% 108% 100% Class D −0.41%−1.83% −1.92% 107% 101% Class F −0.65% −1.14% −1.46% 108% 101%

TABLE 29 Low delay B Main10 Over VTM-3.0 Y U V EncT DecT Class A1 ClassA2 Class B −0.20% −1.05% −0.92% 104%  96% Class C −0.20% −0.53% −0.43%106% 100% Class E −0.26% −0.65% −1.30% 102% 101% Overall −0.21% −0.78%−0.85% 104%  98% Class D −0.11% −0.26% −0.59% 105% 100% Class F −0.18%−0.23% −0.03% 105%  99%

TABLE 30 Random access Main10 Over VTM-3.0 Y U V EncT DecT Class A1−1.28% −2.04% −2.29% 108%  98% Class A2 −0.68% −1.74% −1.33% 104%  98%Class B −0.67% −2.47% −3.16% 106% 100% Class C −0.70% −1.18% −1.84% 105%100% Class E Overall −0.80% −1.89% −2.27% 106%  99% Class D −0.35%−1.64% −1.81% 106% 101% Class F −0.65% −1.21% −1.45% 107% 100%

TABLE 31 Low delay B Main10 Over VTM-3.0 Y U V EncT DecT Class A1 ClassA2 Class B −0.24% −0.59% −1.16% 103%  96% Class C −0.23% −0.95% −0.60%104% 100% Class E −0.27% −0.40% −2.10% 101% 100% Overall −0.24% −0.66%−1.21% 103%  98% Class D −0.11% −0.49% −0.06% 103% 100% Class F −0.27%−0.33% −0.14% 103%  98%

Table 32 to Table 36 show the result of Test 3 in which features (A),(B), and (C) are combined. Table 32 to Table 34 show the result of thetest in which an inter MTS is off, and Table 35 and Table 36 show theresult of the test in which the inter MTS is on.

TABLE 32 All Intra Main10 Over VTM-3.0 Y U V EncT DecT Class A1 −1.88%−3.12% −2.93% 125% 94% Class A2 −0.78% −2.11% −1.64% 128% 96% Class B−0.96% −2.49% −3.30% 125% 97% Class C −1.10% −1.87% −2.49% 124% 99%Class E −1.76% −3.01% −3.91% 124% 98% Overall −1.25% −2.48% −2.88% 125%97% Class D −0.67% −1.65% −1.87% 123% 100%  Class F −1.10% −1.59% −1.87%124% 100% 

TABLE 33 Random access Main10 Over VTM-3.0 Y U V EncT DecT Class A1−1.21% −1.85% −2.19% 109%  99% Class A2 −0.62% −1.32% −1.13% 106%  99%Class B −0.59% −1.99% −3.04% 107% 101% Class C −0.48% −1.08% −1.34% 107%100% Class E Overall −0.69% −1.59% −2.03% 107% 100% Class D −0.26%−1.34% −1.34% 106% 100% Class F −0.51% −0.92% −1.10% 107% 101%

TABLE 34 Low delay B Main10 Over VTM-3.0 Y U V EncT DecT Class A1 ClassA2 Class B −0.22% −1.10% −1.19% 104%  98% Class C −0.15% −0.20% −0.36%105% 100% Class E −0.17% −0.61% −1.96% 101% 101% Overall −0.18% −0.68%−1.11% 103% 100% Class D −0.10% −0.17% −0.43% 104% 101% Class F −0.16%−0.38% −0.96% 104%  99%

TABLE 35 Random access Main10 Over VTM-3.0 Y U V EncT DecT Class A1−1.26% −1.78% −2.19% 107% 98% Class A2 −0.65% −1.41% −1.17% 104% 98%Class B −0.63% −2.10% −3.00% 105% 100%  Clsss C −0.55% −1.03% −1.35%105% 99% Class E Overall −0.74% −1.61% −2.03% 105% 99% Class D −0.29%−1.60% −1.35% 105% 99% Class F −0.53% −0.98% −1.12% 105% 100% 

TABLE 36 Low delay B Main10 Over VTM-3.0 Y U V EncT DecT Class A1 ClassA2 Class B −0.23% −0.55% −1.04% 103%  99% Class C −0.20% −0.62% −0.30%103% 100% Class E −0.24% −0.82% −2.70% 101% 100% Overall −0.22% −0.64%−1.21% 102% 100% Class D −0.10% −0.52% −0.78% 102%  96% Class F −0.28%−0.51% −0.51% 103%  99%

Table 37 to Table 39 show the result of Test 4 in which features (A),(B), and (D) are combined. Table 37 to Table 39 show the result of thetest in which an inter MTS is off

TABLE 37 All Intra Main10 Over VTM-3.0 Y U V EncT DecT Class A1 −1.89%−3.01% −2.80% 128% 94% Class A2 −0.80% −2.18% −1.70% 132% 96% Class B−1.03% −2.49% −3.33% 129% 97% Class C −1.30% −2.07% −3.05% 128% 100% Class E −1.90% −2.98% −3.60% 128% 97% Overall −1.34% −2.50% −2.95% 129%97% Class D −0.97% −2.15% −2.63% 128% 102%  Class F −1.24% −1.78% −2.08%130% 100% 

TABLE 38 Random access Main10 Over VTM-3.0 Y U V EncT DecT Class A1−1.20% −1.76% −1.99% 109%  98% Class A2 −0.60% −1.44% −1.18% 106%  99%Class B −0.60% −1.84% −2.98% 107% 100% Clsss C −0.55% −1.31% −1.49% 107%100% Class E Overall −0.71% −1.60% −2.02% 107% 100% Class D −0.39%−1.64% −1.77% 107% 100% Class F −0.59% −1.19% −1.58% 108% 101%

TABLE 39 Low delay B Main10 Over VTM-3.0 Y U V EncT DecT Class A1 ClassA2 Class B −0.18% −0.84% −1.16% 104%  98% Class C −0.14% −0.47% −0.58%105% 100% Class E −0.21% −0.65% −1.30% 102% 100% Overall −0.18% −0.35%−1.00% 104%  99% Class D −0.11% −0.46% −0.60% 105% 101% Class F −0.22%−0.70% −0.97% 105% 100%

Complexities of Test 1 to Text 4 are shown in the following table.

TABLE 40 Item Test 1 Test 2 Operation counts Primary transform: Primarytransform: Same as VTM Same as VTM Secondary transform: Secondarytransform: RST4 × 4: 256 (M), RST4 × 4: 256 (M), 240(A), 16(S) 240(A),16(S) RST8 × 8: 1024 (M), RST8 × 8: 1024 (M), 960(A), 64(S) 960(A),64(S) Worst Case Worst Case multiplication per multiplication persample: 16 sample: 8 Memory requirements RST8 × 8: RST8 × 8:16*64*8/(8*1024) = 1 16*64*8/(8*1024) = 1 KB per kernel KB per kernelRST4 × 4: RST4 × 4: 16*16*8/(8*1024) = 16*16*8/(8*1024) = 0.25 KB perKernel 0.25 KB per Kernel Total = 2*4*(1 + 0.25) = Total = 2*4*(1 +0.25) = 10 KB 10 KB Bit-precision to represent transform 8 bits 8 bitscoefficients Precision of arithmetic operations during Same as VTM Sameas VTM transform computation Bit-precisions for storing intermediatedata Same as VTM Same as VTM representation Specify if a transformrequires multiple iterations No No where transform output is fed back asinput to the transform logic, and multipie iterations are required toproduce final transform output. If yes, report the number of arithmeticoperation required in each iteration, and if they can be computed inparallel. Other operations and memory requirements N/A N/A List of allcombinations of transform types Two RST Kernel Two RST Kernel combinedwith primary combined with transform in VTM primary transform in VTMList of all combinations of transforms block size and For blocks withFor blocks with min(W, H) < 8: RST4 × 4 transform type used forsecondary transformation min(W, H) < 8: RST4 × 4 For blocks with min(W,H) >= 8: RST8 × 8 For blocks with For simplification (option (B), 8min(W, H) >= 8: RST8 × 8 mult/sample): for 4 × 4 block: Half RST4 × 4for 8 × 8 block: Half RST8 × 8 for 4*8 and 8*4: only apply on one 4 × 4corner Provide analysis of implementation and arithmetic Primarytransform: Primary transform: Same as VTM commonalities of proposedtransforms. {e.g., Same as VTM Secondary transform: Reduced matrixstating which transforms are implemented using Secondary transform:multiplications matrix multiplications, addition-only butterflies, orReduced matrix Givens rotations). multiplications If the proposalrequires additional computations at N/A N/A the encoder or decoder(e.g., additional transforms, or partial transforms along blockboundaries), then the additional operations should be reported togetherwith the other complexity metrics. Operation counts Primary transform:Same as VTM Primary transform: Same as VTM Secondary transform: RST4 ×4: 256 Secondary transform: RST4 × 4: (M), 240(A), 16(S) 256 (M),240(A), 16(S) RST8 × 8: 1024 (M), 960(A), 64(S) RST8 × 8: 768 (M),720(A), 48(S) Worst Case multiplication per Worst Case multiplicationper sample: 8 sample: 8 Memory requirements RST8 × 8: RST8 × 8:16*64*8/(8*1024) = 1 KB per 16*48*8/(8*1024) = 0.75 KB kernel per kernelRST4 × 4: RST4 × 4: 16*16*8/(8*1024) = 0.25 KB per 16*16*8/(8*1024) =0.25 KB per Kernel Kernel Total = 2*4*(1 + 0.25) = 10 KB Total =2*4*(0.75 + 0.25) = 8 KB Bit-precision to represent transform 8 bits 8bits coefficients Precision of arithmetic operations during Same as VTMSame as VTM transform computation Bit-precision for storing intermediatedata Same as VTM Same as VTM representation Specify if a transformrequires multiple No No iterations where transform output is fed back asinput to the transform logic, and muhiple iterations are required toproduce final transform output. if yes, report the number of arithmeticoperation required in each iteration, and if they can be computed inparallel. Other operations and memory N/A N/A requirements List of allcombinations of transform types Two RST Kernel combined with Two RSTKernel combined with primary transform in VTM primary transform in VTMList of all combinations of transforms For blocks with min(W, H) < 8:For blocks with min(W, H) < 8: block size and transform type used forRST4 × 4 RST4 × 4 secondary transformation For blocks with min(W, H) >=8: For blocks with min(W, H) >= 8: RST8 × 8 RST8 × 8 For simplification(option {B}, 8 For simplification (option {B}, 8 mult/sample):mult/sample): for 4 × 4 block: No RST for 4 × 4 block: Half RST4 × 4 for8 × 8 block: Half RST8 × 8 for 8 × 8 block: Half RST8 × 8 for 4*8 and8*4: only apply on one for 4*8 and 8*4: only apply on 4 × 4 corner one 4× 4 corner Provide analysis of implementation and Primary transform:Same as VTM Primary transform: Same as VTM arithmetic commonalities ofproposed Secondary transform: Reduced Secondary transform: Reducedtransforms. (e.g., stating which transforms matrix multiplicationsmatrix multiplications are implemented using matrix multiplications,addition-only butterflies, or Givens rotations). If the proposalrequires additional N/A N/A computations at the encoder or decoder(e.g., additional transforms, or partial transforms along blockboundaries), then the additional operations should be reported togetherwith the other complexity metrics.

As shown in Test 1 to Test 4, applying the RST using the four transformsets, each of which includes two RST candidates, without MTS signalingimproves BD rate, encoding time, and decoding time respectively to−1.59%/128%/97% (AI), −0.88%/108%/100% (RA), and −0.24%/104%/101% (LD).Further, by limiting the maximum number of multiplications per top pixelin a region where the transform is applied, BD rate, encoding time, anddecoding time are improved to −1.40%/128%/97% (AI), −0.76%/108%/100%(RA), and −0.21%/104%/98% (LD), respectively. That is, a significantimprovement in BD rate is achieved with a reasonable increase in memorycosts and encoding time.

An example of a secondary transform process applied to the aboveembodiments is shown in the following tables.

TABLE 41 7.3.2.1 Sequence parameter set RBSP syntax Descriptorseq_parameter_set_rbsp( ) {  ...... u(1)  sps_mts_intra_enabled_flagu(1)  sps_mts_inter_enabled_flag u(1)  sps_st_enabled_flag u(1)  ......}

7.4.3.1 Sequence Parameter Set RBSP Semantics

sps_st_enabled_flag equal to 1 specifies that st_idx may be present inthe residual coding syntax for intra coding untis, sps_st_enabled_flagequal to 0 specifies that st_idx is not present in the residual codingsyntax for intra coding untis.

TABLE 42 7.3.4.12 Residual coding syntax Descriptor residual_coding( x0,y0, log2TbWidth, log2TbHeight, cIdx ) {  if( transform_skip_enabled_flag&& ( cIdx ! = 0 | | tu_mts_flag[ x0 ][ y0 ] = = 0 ) &&   ( log2TbWidth<= 2 ) && ( log2TbHeight <= 2 ) )   transform_skip_flag[ x0 ][ y0 ][cIdx ] ae(v)  last_sig_coeff_x_prefix ae(v)  last_sig_coeff_y_prefixae(v)  if( last_sig_coeff_x_prefix > 3 )   last_sig_coeff_x_suffix ae(v) if( last_sig_coeff_y_prefix > 3 )   last_sig_coeff_y_suffix ae(v) log2SbSize = ( Min( log2TbWidth, log2TbHeight ) < 2 ? 1 : 2 ) numSbCoeff = 1 << ( log2SbSize << 1 )  lastScanPos = numSbCoeff lastSubBlock = ( 1 << ( log2TbWidth + log2TbHeight − 2 * log2SbSize ) )− 1  do {   if( lastScanPos = = 0 ) {    lastScanPos = numSbCoeff   lastSubBlock− −   }   lastScanPos− −   xS = DiagScanOrder[log2TbWidth − log2SbSize ][ log2TbHeight − log2SbSize ]         [lastSubBlock ][ 0 ]   yS = DiagScanOrder[ log2TbWidth − log2SbSize ][log2TbHeight − log2SbSize ]         [ lastSubBlock ][ 1 ]   xC = ( xS <<log2SbSize ) +     DiagScanOrder[ log2SbSize ][ log2SbSize ][lastScanPos ][ 0 ]   yC = ( yS << log2SbSize ) +     DiagScanOrder[log2SbSize ][ log2SbSize ][ lastScanPos ][ 1 ]  } while( ( xC !=LastSignificantCoeffX ) | | ( yC != LastSignificantCoeffY ) ) numSigCoeff = 0  QState = 0  for( i = lastSubBlock; i >= 0; i− − ) {  startQStateSb = QState   xS = DiagScanOrder[ log2TbWidth − log2SbSize][ log2TbHeight − log2SbSize ]         [ lastSubBlock ][ 0 ]   yS =DiagScanOrder[ log2TbWidth − log2SbSize ][ log2TbHeight − log2SbSize ]        [ lastSubBlock ][ 1 ]   inferSbDcSigCoeffFlag = 0   if( ( i <lastSubBlock ) && ( i > 0 ) ) {    coded_sub_block_flag[ xS ][ yS ]ae(v)    inferSbDcSigCoeffFlag = 1   }   firstSigScanPosSb = numSbCoeff  lastSigScanPosSb = −1   remBinsPass1 = ( log2SbSize < 2 ? 6 : 28 )  remBinsPass2 = ( log2SbSize < 2 ? 2 : 4 )   firstPosMode0 = ( i = =lastSubBlock ? lastScanPos − 1 : numSbCoeff − 1 )   firstPosMode1 = −1  firstPosMode2 = −1   for( n = ( i = = firstPosMode0; n >= 0 &&remBinsPass1 >= 3; n− − ) {    xC = ( xS << log2SbSize ) +DiagScanOrder[ log2SbSize ][ log2SbSize ][ n ][ 0 ]    yC = ( yS <<log2SbSize ) + DiagScanOrder[ log2SbSize ][ log2SbSize ][ n ][ 1 ]   if( coded_sub_block_flag[ xS ][ yS ] && (n > 0 | |!inferSbDcSigCoeffFlag ) ) {     sig_coeff_flag[ xC ][ yC ] ae(v)    remBinsPass1− −     if( sig_coeff_flag[ xC ][ yC ] )     inferSbDcSigCoeffFlag = 0    }    if( sig_coeff_flag[ xC ][ yC ] ){     numSigCoeff++     if( ( ( ( log2TbWidth == 2 && log2TbHeight == 2) | | ( log2TbWidth == 3 &&      log2TbHeight == 3 ) ) && n >= 8 ) | | (xC >= 4 && yC >= 4 ) ) {      if( cIdx != 0 ) {      numZeroOutSigCoeffC++      } else {       numZeroOutSigCoeff++     }     }     abs_level_gt1_flag[ n ] ae(v)     remBinsPass1− −    if( abs_level_gt1_flag[ n ] ) {      par_level_flag[ n ] ae(v)     remBinsPass1− −      if( remBinsPass2 > 0 ) {       remBinsPass2− −      if( remBinsPass2 = = 0 )        firstPosMode1 = n − 1      }     }    if( lastSigScanPosSb = = −1 )      lastSigScanPosSb = n    firstSigScanPosSb = n    }    AbsLevelPass1[ xC ][ yC ] =     sig_coeff_flag[ xC ][ yC ] + par_level_flag[ n ] +abs_level_gt1_flag[ n ]    if( dep_quant_enabled_flag )     QState =QStateTransTable[ QState ][ AbsLevelPass1[ xC ][ yC ] & 1 ]    if(remBinsPass1 < 3 )     firstPosMode2 = n − 1   }   if( firstPosMode1 <firstPosMode2 )    firstPosMode1 = firstPosMode2   for( n = numSbCoeff −1; n >= firstPosMode2; n− − )    if( abs_level_ gt1_flag[ n ] )    abs_level_gt3_flag[ n ] ae(v)   for( n = numSbCoeff− 1; n >=firstPosMode1; n− − ) {    xC = ( xS << log2SbSize ) + DiagScanOrder[log2SbSize ][ log2SbSize ][ n ][ 0 ]    yC = ( yS << log2SbSize ) +DiagScanOrder[ log2SbSize ][ log2SbSize ][ n ][ 1 ]    if(abs_level_gt3_flag[ n ] )     abs_remainder[ n ] ae(v)    AbsLevel[ xC][ yC ] = AbsLevelPass1[ xC ][ yC ] +           2 * (abs_level_gt3_flag[ n ] + abs_remainder [ n ] )   }   for( n =firstPosMode1; n > firstPosMode2; n− − ) {    xC = (xS << log2SbSize ) +DiagScanOrder[ log2SbSize ][ log2SbSize ][ n ][ 0 ]    yC = ( yS <<log2SbSize ) + DiagScanOrder[ log2SbSize ][ log2SbSize ][ n ][ 1 ]   if( abs_level_gt1_flag[ n ] }     abs_remainder[ n ] ae(v)   AbsLevel[ xC ][ yC ] = AbsLevelPass1[ xC ][ yC ] + 2 * abs_remainder[n ]   }   for( n = firstPosMode2; n >= 0; n− − ) {    xC = ( xS <<log2SbSize ) + DiagScanOrder[ log2SbSize ][ log2SbSize ][ n ][ 0 ]    yC= ( yS << log2SbSize ) + DiagScanOrder[ log2SbSize ][ log2SbSize ][ n ][1 ]    dec_abs_level[ n ] ae(v)    if(AbsLevel[ xC ][ yC ] > 0 )    firstSigScanPosSb = n    if( dep_quant_enabled_flag )     QState =QStateTransTable[ QState ][ AbsLevel[ xC ][ yC ] & 1 ]   }   if(dep_quant_enabled_flag | | !sign_data_hiding_enabled_flag )   signHidden = 0   else    signHidden = ( lastSigScanPosSb −firstSigScanPosSb > 3 ? 1 : 0 )   for( n = numSbCoeff− 1; n >= 0; n− − ){    xC = ( xS << log2SbSize ) + DiagScanOrder[ log2SbSize ][ log2SbSize][ n ][ 0 ]    yC = ( yS << log2SbSize ) + DiagScanOrder[ log2SbSize ][log2SbSize ][ n ][ 1 ]    if( sig_coeff_flag[ xC ][ yC ] &&     (1signHidden | | ( n != firstSigScanPosSb ) ) )     coeff_sign_flag[ n ]ae(v)   }   if( dep_quant_enabled_flag ) {    QState = startQStateSb   for( n = numSbCoeff− 1; n >= 0; n− − ) {     xC = ( xS << log2SbSize) +       DiagScanOrder[ log2SbSize ][ log2SbSize ][ n ][ 0 ]     yC = (yS << log2SbSize ) +       DiagScanOrder[ log2SbSize ][ log2SbSize ][ n][ 1 ]     if( sig_coeff_flag[ xC ][ yC ] ) {      TransCoeffLevel[ x0][ y0 ][ cIdx ][ xC ][ yC ] =        ( 2 * AbsLevel[ xC ][ yC ] − (QState > 1 ? 1 : 0 ) ) *        ( 1 − 2 * coeff_sign_flag[ n ] )    QState = QStateTransTable[ QState ][ par_level_flag[ n ] ]  } else {  sumAbsLevel = 0    for( n = numSbCoeff − 1; n >= 0; n− − ) {     xC =( xS << log2SbSize ) +       DiagScanOrder[ log2SbSize ][ log2SbSize ][n ][ 0 ]     yC = ( yS << log2SbSize ) +       DiagScanOrder[ log2SbSize][ log2SbSize ][ n ][ 1 ]     if( sig_coeff_flag[ xC ][ yC ] ) {     TransCoeffLevel[ x0 ][ y0 ][ cIdx ][ xC ][ yC ] =        AbsLevel[xC ][ yC ] * ( 1 − 2 * coeff_sign_flag[ n ] )      if( signHidden ) {      sumAbsLevel += AbsLevel[ xC ][ yC ]       if( ( n = =firstSigScanPosSb ) && ( sumAbsLevel % 2 ) = = 1 ) )       TransCoeffLevel[ x0 ][ y0 ][ cIdx ][ xC ][ yC ] =         −TransCoeffLevel[ x0 ][ y0 ][ cIdx ][ xC ][ yC ]      }     }   }   }  }  if( tu_mts_flag[ x0 ][ y0 ] && ( cIdx = = 0 ) )   mts_idx[x0 ][ y0 ][ cIdx ] ae(v) }

7.3.4.12 Residual Coding Syntax

TABLE 43 7.3.4.6 Coding Unit Syntax Descriptor coding_unit( x0, y0,cbWidth, cbHeight, treeType ) {  if( slice_type != I ) {   cu_skip_flag[x0 ][ y0 ] ae(v)   if( cu_skip_flag[ x0 ][ y0 ] = = 0 )   pred_mode_flag ae(v)  }  if( CuPredMode[ x0 ][ y0 ] = = MODE_INTRA ){   if( pcm_enabled_flag &&    cbWidth >= MinIpcmCbSizeY && cbWidth <=MaxIpcmCbSizeY &&    cbHeight >= MinIpcmCbSizeY && cbHeight <=MaxIpcmCbSizeY )    pcm_flag[ x0 ][ y0 ] ae(v)   if( pcm_flag[ x0 ][ y0] ) {    while( !byte_aligned( ) )     pcm_alignment_zero_bit  f(1)   pcm_sample( cbWidth, cbHeight, treeType)   } else {    if( treeType == SINGLE_TREE | | treeType = = DUAL_TREE_LUMA ) {     if( ( y0 %CtbSizeY ) > 0 )      intra_luma_ref_idx[ x0 ][ y0 ] ae(v)     if(intra_luma_ref_idx[ x0 ][ y0 ] = = 0)      intra_luma_mpm_flag[ x0 ][y0 ] ae(v)     if( intra_luma_mpm_flag[ x0 ][ y0 ] )     intra_luma_mpm_idx[ x0 ] [ y0 ] ae(v)     else     intra_luma_mpm_remainder[ x0 ] [ y0 ] ae(v)    }    if( treeType == SINGLE_TREE | | treeType = = DUAL_TREE_CHROMA )    intra_chroma_pred_mode[ x0 ] [ y0 ] ae(v)   }  } else { /*MODE_INTER */   if( cu_skip_flag[ x0 ][ y0 ] = = 0 ) {    merge_flag[ x0][ y0 ] ae(v)   if( merge_flag[ x0 ][ y0 ] ) {    merge_data( x0, y0,cbWidth, cbHeight )   } else {    if( slice_type = = B )    inter_pred_idc[ x0 ][ y0 ] ae(v)    if( sps_affine_enabled_flag &&cbWidth >= 16 && cbHeight >= 16 ) {     inter_affine_flag[ x0 ][ y0 ]ae(v)     if( sps_affine_type_flag && inter_affine_flag[ x0 ][ y0 ] )     cu_affine_type_flag[ x0 ][ y0 ] ae(v)    }    if( inter_pred_idc[x0 ][ y0 ] != PRED_L1 ) {     if( num_ref_idx_l0_active_minus1 > 0 )     ref_idx_l0[ x0 ][ y0 ] ae(v)     mvd_coding( x0, y0, 0, 0 )     if(MotionModelIdc[ x0 ][ y0 ] > 0 )      mvd_coding( x0, y0, 0, 1 )    if(MotionModelIdc[ x0 ][ y0 ] > 1 )      mvd_coding( x0, y0, 0, 2 )    mvp_l0_flag[ x0 ][ y0 ] ae(v)    } else {     MvdL0[ x0 ][ y0 ][ 0 ]= 0     MvdL0[ x0 ][ y0 ][ 1 ] = 0    }    if( inter_pred_idc[ x0 ][ y0] != PRED_L0 ) {     if( num_ref_idx_l1_active_minus1 > 0 )     ref_idx_l1[ x0 ][ y0 ] ae(v)     if( mvd_l1_zero_flag &&inter_pred_idc[ x0 ][ y0 ] = = PRED_BI ) {      MvdL1[ x0 ][ y0 ][ 0 ] =0      MvdL1[ x0 ][ y0 ][ 1 ] = 0      MvdCpL1[ x0 ][ y0 ][ 0 ][ 0 ] = 0     MvdCpL1[ x0 ][ y0 ][ 0 ][ 1 ] = 0      MvdCpL1[ x0 ][ y0 ][ 1 ][ 0] = 0      MvdCpL1[ x0 ][ y0 ][ 1 ][ 1 ] = 0      MvdCpL1[ x0 ][ y0 ][ 2][ 0 ] = 0      MvdCpL1[ x0 ][ y0 ][ 2 ][ 1 ] = 0     } else {     mvd_coding( x0, y0, 1, 0)     if( MotionModelIdc[ x0 ][ y0 ] > 0 )     mvd_coding( x0, y0, 1, 1 )     if(MotionModelIdc[ x0 ][ y0 ] > 1 )     mvd_coding( x0, y0, 1, 2 )     mvp_l1_flag[ x0 ][ y0 ] ae(v)    }else {     MvdL1[ x0 ][ y0 ][ 0 ] = 0     MvdL1[ x0 ][ y0 ][ 1 ] = 0   }     if( sps_amvr_enabled_flag && inter_affine_flag = = 0 &&      (MvdL0[ x0 ][ y0 ][ 0 ] != 0 | | MvdL0[ x0 ][ y0 ][ 1 ] != 0 | |      MvdL1[ x0 ][ y0 ][ 0 ] != 0 | | MvdL1[ x0 ][ y0 ][ 1 ] != 0 ) )     amvr_mode[ x0 ][ y0 ] ae(v)      if( sps_gbi_enabled_flag &&inter_pred_idc[ x0 ][ y0 ] = = PRED_BI &&       cbWidth * cbHeight >=256 )      gbi_idx[ x0 ][ y0 ] ae(v)    }  }  if( !pcm_flag[ x0 ][ y0 ]) {    if( CuPredMode[ x0 ][ y0 ] != MODE_INTRA && cu_skip_flag[ x0 ][y0 ] = = 0 )     cu_cbf ae(v)    if( cu_cbf ) {     if( treeType ==SINGLE_TREE ) {      numZeroOutSigCoeff = 0      numZeroOutSigCoeffC = 0    } else if( treeType == DUAL_TREE_LUMA ) {      numZeroOutSigCoeff =0     } else {      numZeroOutSigCoeffC = 0     }     transform_tree(x0, y0, cbWidth, cbHeight, treeType )     if( ( treeType == SINGLE_TREE| | cIdx == 0 | | Min(cbWidth, cbHeight ) < 4 ) &&     sps_st_enabled_flag == 1 && CuPredMode[ x0 ][ y0 ] = = MODE_INTRA ){      if( ( numSigCoeff > ( ( treeType == SINGLE_TREE ) ? 2 : 1 ) ) &&      ( ( treeType == SINGLE_TREE && numZeroOutSigCoeff == 0 &&       numZeroOutSigCoeffC == 0 ) | | ( treeType == DUAL_TREE_LUMA &&       numZeroOutSigCoeff == 0 ) | | ( treeType == DUAL_TREE_CHROMA &&       numZeroOutSigCoeffC == 0 ) ) ) ) {       st_idx[ x0 ][ y0 ]     }     }    }  } }

7.4.5.6 Coding Unit Semantics

st_idx[x0] [y0] specifies which secondary transform kernel is appliedbetween two candidate kernels in a selected transform set. st_idx[x0][y0] equal to 0 specifies that the secondary transform is not applied.The array indices x0, y0 specify the location [x0, y0] of the top-leftsample of the conisdered transform block relative to the top-left sampleof the picture.

When st_idx[x0] [y0] is not present, st_idx[x0] [x0] [y0] is inferred tobe equal to 0.

TABLE 44

indicates data missing or illegible when filed

8.5.4. Transformation process far scaled transform coefficients

8.5.4.1 General

Inputs to this process are:

-   -   a luma location (xTbY, yTbY) specifying the top-left samle of        the current luma transform block relative to the top-lieft luma        sample of the current picture,    -   a variable nTbW specifying the width of the current transform        block,    -   a variable nTbH specifying the height of the current transform        block,    -   a variable cIdx specifying the colourr component of the current        block,    -   an (nTbW)×(nTbH) array d[x] [y] of scaled transform coefficients        with x=0..sTbW−1, y=0..nTbH−1.

Output of this process is the (nTbW)×(nTbH) aray r[x] [y] of residualsamples with x=0..nTbW−1, y=0. .nTbH−1.

If st_idx[xTbY] [yTbY] [cIdx] is not equal to 0, the following applies:

-   -   1. The variables nStSize, log2StSize, numStX, numStY, and        nonZeroSize are derived as follows:        -   If both nTbW and nTbH are greater than or equal to 8,            nStSize is set to 8 and log2StSize is set to 3.        -   Otherwise, nStSize is set to 4 and log2StSize is set to 2.        -   If nTbH is equal to 4 and nTbW is greater than 8, numStX set            equal to 2.        -   Otherwise, numStX set equal to 1.        -   If nTbW is equal to 4 and nTbH is greater than 8, numStY set            equal to 2.        -   Otherwise, numStY set equal to 1.        -   If both nTbW and nTbH are equal to 4 or both nTbW and nTbH            are equal to 8, nonZeroSize is set equal to 8.        -   Otherwise, nonZeroSize set equal to 16.    -   2. For xShIdx=0..numStX−1and ySbIdx=0..numStY−1, the following        applies:        -   The variable array u[x] with x=0..nonZeroSize−1 are derived            as follows;            -   xC=(xSbIdx<<log2StSize)+DiagScanOrder[log2StSize]                [log2StSize] [x] [0]            -   yC=(ySbIdx<<log2StSize)+DiagScanOrder[log2StSize]                [log2StSize] [x] [1]            -   u[x]=d[xC] [yC]        -   u[x] with x=0..nonZeroSize−1 is transformed to the variable            array v[x] with x=0..(1<<(log2StSize<<1))−1 by invoking the            one-dimensional transformation process as specified in            claused 8.5.4.4 with the transform input length of the            scaled transform coefficients nonZeroSize, the transform            output length (1<<(log2StSize<<1))−1, the list u[x] with            x=0..nonZeroSize−1, the index for transform set selection            stPredModeIntra, and the index for transform selection in a            transform set st_idx[xTbY] [xTbY] [cIdx] as inputs, and the            output is the list v[x] with x=0..(1<<(log2StSize<<1))−1.            The variable stPredModeIntra is set to the predModeIntra            specified in clause 8.2.4.2.1.        -   The array d[(xSbIdx<<log2StSize)+x] [(ySbIdx<<log2StSize)+y]            with x=0..nStSize−1, y=0..nStSize−1 are derived as follows:            -   If stPredModeIntra is less than or equal to 34, or equal                to INTRA_LT_CCLM, INTRA_T_CCLM, or INTRA_L_CCLM, the                following applies:                -   d[(xSbIdx<<log 2StSize)+x]                    [ySbIdx<<log2StSize)+y]=v[x+(y<<log2StSize)]            -   Otherwise, the following applies:                -   d[(xSbIdx<<log2StSize)+x]                    [(ySbIdx<<log2StSize)+y]=v[y+(x<<log2StSize)]

The varlable trTypeHor specifying the horizontal transform kernel andthe variable trTypeVer specifying the vertical transform kernel arederivedin Table 8-16 depending on mts_idx[xTbY] [yTbY] [cIdx].

The variables nonZeroW and nonZeroH are derived as follows:

nonZeroW=Min(nTbW, 32)   (8-833)

nonZeroH=Min(nTbH, 32)   (8-834)

The (nTbW|x(nTbH) array r of residual samples is derived as follows:

-   -   4. Each (vertical) column of scaled transform coefficients d[x]        [y] with x=0..nonZeroW−1, y=0..nonZeroH−1 is transformed to e[x]        [y] with x=0..nonZeroW−1, y=0..nTbH−1 by working the        one-dimensional transformation process as specified in clause        8.5.4.2 for each column x=0..nonZeroW−1 with the height of the        transform block nTbH, the non-zero height of the scaled        transform coefficients nonZeroH, the list d[x] [y] with        y=0..nonZeroH−1 and the transform type variable trType set equal        to trTypeVer as inputs, and the output is the list e[x] [y] with        y=0..nTbH−1.    -   5. The intermediate sample values g[x] [y] with x=0..nonZeroW−1,        y=0..nTbH−1 are derived as follows:

g[x] [y]=Clip3(CoeffMin, CoeffMax, (e[x] [y]+64)>>7)   (8-835)

-   -   6. Each (horizontal) row of the resulting array g[x] [y] with        x=0..nonZeroW−1, y=0..nTbH−1 is transformed to r[x] [y] with        x=0..nTbW−1, y=0..nTbH−1 by invoking the one-dimensional        transformation process as specified in clause 8.5.4.2 for each        row y=0..nTbH−1 with the width of the tranform block nTbW, the        non-zero width of the resulting array g[x] [y] nonZeroW, the        list g[x] [y] with x=0..nonZeroW−1 and the transform type        variable trType set equal to trTypeHor as inputs, and the output        is the list r[x] [y] with x=0..nTbW−1.

TABLE 8-16 Specification of trTypeHor and trTypeVer depending onmts_idx[ x ][ y ][ cIdx ] mts_idx[ xTbY ][ yTbY ][ cIdx ] trTypeHortrTypeVer −1 0 0 0 1 1 1 2 1 2 1 2 3 2 2

8.5.4.2 Primary Transformation Process

8.5.4.3 Primary Transformation Matrix Derivation Process

8.5.4.4 Secondary Transformation Process

Inputs to this: process are:

-   a variable nTrS specifying the transform output length,-   a variable nonZeroSize specifying the transform input length,-   a list of transform input x[j] with j=0..nonZeroSize−1,-   a variable stPredModeIntra specifying the index for transform set    selection,-   a variable stIdx specifying the index for transform selection in a    set. Output of this process is the list of transformed samples y[i]    with i=0..nTrS−1.

The transtormation matrix derivation process as specified in clause8.5.4.5 is involved with the transform output length nTrS, the index fortransform set selection stPredModeIntra, and the index for transformselection in a transform set stldx as iuputs, and the transformationmatrix secTransM atrix as output.

The list of transformed samples y[i] with i=0..nTrS−1 is derived asfollows:

y[i]=Σ_(i=0) ^(nonZeroSize−1)secTransMatrix[l] [i] *x[l] withi=0..nTrS−1

8.5.4.5 Secondary Traasformation Matrix Derivation Process

Inputs to this Process are:

-   a variable nTrS specifying the transform output length,-   a variable stPredModeIntra specifying the index for transform set    selection,-   a variable stIdx specifying the index for transform selection in the    designated transform set. Output of this process is the    transformation matrix secTransMatrix.

The variable stTrSetIdx is derived as follows:

stPredModeIntra stTrSetIdx stPredModeIntra < 0 1 0 <= stPredModeIntra <=1 0 2 <= stPredModeIntra <= 12 1 13 <= stPredModeIntra <= 23 2 24 <=stPredModeIntra <= 44 3 45 <= stPredModeIntra <= 55 2 56 <=stPredModeIntra 1

The transformation atrix secTransMatrix is derived based on nTrS,stTrSetIdx, and stIdx as follows:

TABLE 45 - If nTrS is equal to 16, stTrSetIdx is equal to 0, and stIdxis equal to 1, the following applies: secTransMatrix [ m ][ n ] = { {106 42 31

7 −25 7 0 17 −9 −16

−2

0 }, { −45 99 −19 −37 −13 2 7 31 28 −9

15

}, { −13

−9 −94 26 12

−9

−32

−6 11

−2 }, { 2 −11 4 56

−2 0

2

−11 −15 }, { −45 −2 101 −4 −17 −37

−12

−7

1

−1 }, { 21 −25 −18 −14 −105 12

−10 1

29 5 7 −6 }, { 6 15 −29 29

14 97 −27 −17

16

7 25

}, { −1 2 1 −10 22 −2 −40 4 −28 9

107 −6 15 −16

}, { −10 −12 −54 6 −4 −96 −7 4

−56 −1 −6 11 10

}, { 6 −22 16 18 37 4 12 100 −13 −25

43 29

−9 }, { 1 11 14 18 −25 27

−15 −98

32

}, { −1 3 −2 −13 −4 1 12

−24

−4

−30 −46 }, { 0 1 −6 2 4 59

7

15 −104 2 −10 4 41 −7 }, { −1

1 1 16 −9 −14

5 7

−2 104 −6 44 −27 }, { 0 −1

−4 −5 −19 −12

4

41

−42 14 98 −21 }, { 0 0 0 1 −5 1 9 9 4

−28 −17

−17 −107 },

- If nTrS is equal to 16, stTrSetIdx is equal to 0, and stIdx is equalto 2, the following applies: secTransMatrix [ m ][ n ] = { { −117 −31 125 18 7 19

10 4 8 −4

1 1 }, {

−94 69

4 −2 29 19 −16

7 −1 4

0 }, {

33

91

−1

14 2

−1 0 0 }, {

14 −15 −9 −3 −10 4 −5

−2 41

−7

11 }, {

−56 −96 11

27

7

−9 20 −4 −6

0 }, {

3 90 −74 −21

9 −14

2 10 −2 0 −6 }, {

26 34

42

11

−2

14

4 }, { −2

1 −26 17

−5

40 −87

0 −1 26 −32 }, { 16 21 9

69

7

−4 1 0 }, { −2

−9 −11

79

12 −6 −6

1 −27 −17

}, {

−6 −5

−38

14 19 100

−23 2 1

}, { −1 −7 4 0 4 10 −27 −20 24

−21 −74

}, { 4 6 12 −9 −1 −2 0

−2

−32

}, { −2

−1 −16 11 0 −5

−22

−4

}, { 0

−7

−20 6 0 7 24

72

}, { 0 −2 0 7 −4 4

−4

−19

},

- If nTrS is equal to 16, stTrSetIdx is equal to 1, and stIdx is equalto 1, the following applies: secTransMatrix [ m ][ n ] = { {

−51

−11 12 0 −17 −7 7 −2 −3 3 −6 0 1 1 }, { 41 −29 21 −91

−40 −49 0 −13 4 −3

−9 4 −4 0 }, { 4

−2

−9 31 −92

−24

−2 −6 10 }, {

−1

0 −2

−2 −9 −7 24

−55 68 −22 }, { 48

50 45 29

−17 9 −6 −14 2 0 5 −2 0 }, { 11 44 74

−1

−56 −4 −42 −5 −10 1 9 9 4 }, {

21 −31

−46 −4 −45 −52 11

−10 −14 }, { −1 −1

−8 −14

−21 −4 −6 −16

−71 49 }, { 6 22

9

11 19 4 4 5

}, {

−18 −12

−16 14

−64 10 12

}, { −5 −15 −49 1

−10 −22

16

47 24 }, {

−5 4 −11 −26 2 20

6 −20 55 16 −69 }, {

9 1

15

25 95

−6

−3

}, {

−6 −40 −19 −2

−1

59

5 −4 7 }, { 4 −14

−61 −1 −11

4 −15 −71

26

−6 }, { 2 1 10

−5 12

9 −17 45 6

},

- If nTrS is equal to 16, stTrSetIdx is equal to 1, and stIdx is equalto 2, the following applies: secTransMatrix [ m ][ n ] = { {

19

6 12 1

0 −2 −1

}, {

−17 −25

27

−27

14 27 20 −2 −5 7 }, { −7

24 44

11

−10 40

24 27 −22

}, {

−1

−1

−5 −44 4

−45 −19 47

}, { 68

70

47 −3

−7 5 4 5 }, {

74

6

−43

7

−5 47

18 4 10 }, { −6

10

−11

−1

−19

}, { 1

−17

4

−1 20 −77

−45 74 }, { −13 46 −6 −58 26

−6

20

9 −26 −11

2 }, { −7 −46

−17 −1

11 −18

35

32 5 }, { 11 −7

−64 −10 −24

41 26 }, { −1

2

−26

62 10 67 }, {

−27

−6

−21

71 40

10 −52 −19 9 −3 }, {

−1

−69 −44

−1 }, { −2

−4 56 29 7 62

2 41

−40

}, { −1

−20 7

−11

−41

−2

−16 79

},

- If nTrS is equal to 16, stTrSetIdx is equal to 2, and stIdx is equalto 1, the following applies: secTransMatrix [ m ][ n ] = { { −110

10 −21

10

16 1

−2 1 4 −3 }, { 47 11 85 −9 22

−2 28 9

2

−4 4 9 −2 }, { −2 −2 −73 −2

−62

−17 9

−14

−10 7

−5 }, {

1 17 0 5

0 1 −9

−5

−9 }, {

12

−15 7 −4

11 −4

2 }, { 15 −41

9

−26 −49 15

22

4 }, { 0 0 −22 −2

−20 0

26 25

−24

14 }, { 1 −1 4 1

24 1 −1

−10 −20 78 16

}, { 16 34 −8

−12 3 41 28

9 −15

6 7

}, { −7

−9

20

17 −7 }, { 1 −2

−2 28

1

−47 −20 −39

44

}, { 0 0 −3 0

−1

44

14

71

}, { 11

44 3

−1

2 −14

7 }, {

−7

2

−100

2 −45 10 18 }, { 0 −1

−2 −6 6

−27 −1 −10

−54 4 −88 25

}, { 0 0 −1 1 2 −7 0 1 0 −7 0

53

100 },

- If nTrS is equal to 16, stTrSetIdx is equal to 2, and stIdx is equalto 2, the following applies: secTransMatrix [ m ][ n ] = { { −97 61

49 −16 2

15

−7

4

2 0 }, {

34 74 0

18

−4

24 9

4 0 }, {

−47 2

72

−15

−27 −14

27 7 }, {

4

18 −54

4

−77

−22 47 0

}, { 66

77 17

9 7 8 −12 2 −2 4 1 1 }, {

−22

−15 −46

3 }, {

22

−10 54 −34

−29 −4

67 −2 49

}, { 0

0

−26 32 2 4 47 54 7

14

47 }, { −15 −62

27

64

9 −12

−6

}, { 0

−6

48 1

−29 −61

}, {

0 −2 11 −55

−11

17 12 −40

−11 27 }, { −1 −3

−2 11

3 12

−7 0 −9

10 72 }, { 2

−21

−14 90 −42

−4

6

−4

}, {

−6 4 22

−19 −40

−40

−9

}, { −2 −7 −1 1

−34 22

71 −31

}, { 0

2 −1

−2 −2 −7

26 54

},

- If nTrS is equal to 16, stTrSetIdx is equal to 3, and stIdx is equalto 1, the following applies: secTransMatrix [ m ][ n ] = { {

21

29

−10 19

7 −5

1

2

}, {

41

80

45 −18 16

16 −21 −17 1 1 6 11 }, { −3 −17

21 −77 −14 44 41

−22

21 }, { −2 3

4

−6

8

10

77

62

}, {

46

16 2

}, { 22 59

−5

−72 29

15 −2

8

}, { −12

−39

−23 −29

24

−27 −21

}, { 1

11 9 10 −2 40

}, { −21

−46

5

27

−15 6 }, {

−19

−48

−32

}, {

−21

−44

}, { −2

8

0

48 10

}, {

11

−11 −14 −22 7

7 0 }, { 0

−21 20 41

−47

−9 }, { 4 1

19 0

42

−7 49

}, { 0 6

−10 2

−91

−21 11

−65 },

- If nTrS is equal to 16, stTrSetIdx is equal to 3, and stIdx is equalto 2, the following applies: secTransMatrix [ m ][ n ] = { { −96

4

10 11

−2

−1 −1 }, {

42

−29

19 −17

−2

6 0 }, { 1

−20

74 2

−11

}, {

1

5

−17

}, {

−4 14 −3 −1 }, {

1 54 −9

−64 2

−7 6

8 −1 }, {

31

−55

−27 60 −7

}, {

−3

66 2

−27

−15 −62

}, { −20

−26

−31

−6 −71 2

11 −5 −1 }, { 14

−77

10 −10

−70 0 −2 }, { 0

20 −16 −76 44

−6 7

46 14

−13 }, { 0

−30 −6 −4

2 6 76 }, {

4

20

71

26 1 −6

2 }, {

−4

−29

1 42

46

−2 2 }, { 0 −6

−20

−1

−39

−6 11

}, { 0

2

1

0 −1

60 17

},

- If nTrS is equal to 64, stTrSetIdx is equal to 0, and stIdx is equalto 1 the following applies: secTransMatrix [ m ][ n ] = { { −120 −19 −10

10 0 0 4

}, {

2 −7

−6 }, { 20 44

−40

24 2 −11 −3

4 }, {

−1

7

−7

−3

−1

}, { 4

0

−2

−2

0 9 1 0

}, {

−2 0

2 0 0

0 1

−2 0 −24

}, { 2

−1 0

−1 0

1 2 −1 0

0 }, {

−1 0 4 1 0 0 −4 0 0 −1 0 0

0 }, {

−61

−4

0

7 7 −1 6 }, {

1

−14

27

−30

0 3 −1 −4 }, {

−7

73

−4

−7

10 1

}, { 0

0

−4

−16 −8 1

62 79 −5

}, { −1 0 4 4

−2

−1 6

−2 −34 −2 }, { 0 1 0

−2 −1

−4

1

4 1

}, { 0 0 2 2 2 −1 0 1

0 2

−2

}, { 0

0 1 −1 0

−2 1 0 −1 2

}, { 15 32 29

−14

4 −2 44

5 11 −6 −4

}, { 0 1

79

−1

}, {

−7 −2

17 −12 −2 −9 14

71 −6 −6

}, { 0 −1 4 −1

−4

21

0

}, { −2

0 0 −1 −4

−2 −7

4

4

}, { 0

1 0 −1 −1 4

0 −1 0

−1 2 −4 }, { 0 −1 0

−2 0 −1 −1 −1

1 2 1 −1

}, { 0

0 0 −1 −1 2 −1 0 0 0 −1 0

0

}, {

−4 11

−4

5

1

}, { 0

2

10

−19

49

11

}, { 0 −1 −4 2 −1

4

−14

6

}, { 0 0 0

1 0

2 4 −2

0

}, { 0 0 0 0 −1 2 2

2

2

−30 }, { 0 0 0 −1

0

1 −1 −1

−1 6 0

}, { 0 0 0 0

1 0 1 0 −2 0 0 0 0 0

}, { 0

0 0

0 0 0 0 0 1 −1 2 0 0 0 }, {

7 −4 −1 1 −17 2 9

0

}, { 0 2

4 4 −4

−2 0

4

}, {

0

2 2 2 −6

−16 −3

1 7 }, { 0

0 0

0

−6 9

1 41 }, { 0 0 0 0 0 0

1 0 1 4 −1

−4 −1 −7 }, { 0 0 0 0 0 0 0 1 0 0 0 0 0 −1 1 −2 }, { 0

0 0 0 0 0 0 0 0 1 0 0

−1

}, { 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0

}, { 1 −1

−9

0 −1 9

−4 2

0

}, { 0 1 0

0 1

0 −2

−4 −1 −1

}, { 0

0 0 2 −4 −1 0 0 1 −2

14 0 −2 }, { 0 0 0

0 0 0 1 1 0

−2

−3 0 2 }, { 0 0 0 0 0 0 0

0 0 −1 0 −1 0 1 4 }, { 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 }, { 0 0 0 0 0 0 0

0 0 0 0 0 0 0

}, {

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }, {

0 0

0 0

1 2

11 0 0 }, { 0 1

1

0 1 0

2

2 0 0 −2 }, { 0 −1 0

1

−1 1 1

1 −3 −4 0

}, { 0 0 0 0 −1 0 0 1 2 0 −1 1 0 1 0 4 }, { 0 0 0 0 0 0 0

0 0 0 0 0 0 −2

}, { 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }, { 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 }, {

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }, { 1 0

1 −4 0 0 0

0 −1 0

0

}, { 0 0 0

0 0

−1 0 0 −1 2 −1 1 0 0 }, { 0 0 0 0 0

−1 0 0 0 −1 0 1 0

}, { 0 0 0 0 0 0 0

0 0 1 −1 −2 0 0

}, { 0 0 0 0 0 0 0

0 0 0 0 0 0 1

}, { 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }, { 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 }, { 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 },

- If nTrS is equal to 64, stTrSetIdx is equal to 0, and stIdx is equalto 2 the following applies: secTransMatrix [ m ][ n ] = { {

57

7 20 −12 −1

0 −2 1 −4 −1 }, {

−24 5 10 21 −11

−4 −4

−1 }, {

−35

34 −6

2 16 2 2 }, { −1

−18

−15 −51 97

4 −9 −1 −14 20 }, { −1

5 0 −1 −1

1 0

−9

1 }, {

−1 1 −4 −6 −3 −2 −4 2

−6 }, { −1 −2

0 1 1 2

−4 0 0

0 −1 −5 −2 }, { 0

−2

−1

−1 2 0 0 1 0 0

0 }, {

−15

−29

15

−6 −2

−1 }, {

0

−22

−1

4 0 }, {

11 7

−71

2

−1 −1 7 }, {

5

−14

26

−1

11 −19

4 }, {

−1 −4

4

−10 −11 −9

}, { 0 1 −1

0 1

1

2 4 22

}, {

−1 −1 −1 1 1 0 1 0

2 −4 −1

4 }, {

0 0

0 1 2 1 1 1

0

0 2 0 }, { −8

29

−5

21 −2 0 −1

}, {

−19

−28 −1 −4 −70 77

−24 −5 5 −4 }, { 1 14 −1

0

70

61 −13 17 }, { −1

−2 8 4 0

−7 −7 7 −4

6 77 }, { 0 1 1 2

−4

2

−10

37 29 }, { 0 1 0 2 1 0

−2 2 −2 2 0

1 −1 2 }, { 0 1 0 0 1

0 −1 1 −2 0 0 0 −1 −3 1 }, { 0 0 0 1

0

−1 0

1 −1

0 }, {

−7

−13 14

24

0 −15 9 2

}, { 1 4

−19 15 −2 23

−9

1

−1

−1 −24 }, { −1

1

−7 2 29 −16 0 −2

}, { 0 −2 0 11 0

−13

12 −38 1

−4 1

}, { 0 0 1 1 0

−7 −6 0

−2 3

2

}, { 0

0 1 −1 0 −2 1

−2 −1

1 −7

}, { 0 0 0 0 0 0

−1 0

−2 0 5

}, { 0 0 0 0 0 0 −1

0 −1

0 1

0 }, { −1 −7

−1 2

4 −2 0 2 14 1 −1

}, { 1 0 4

0

−11

−21 1

}, { 0 1 −1

0 −3 5

27 0

}, { 0 0 −1

1 0 −7 7 0

−6

}, { 0

0 −1 0

0 −1

0

1

0 }, { 0

0 0 0 0 0 1 0

0 −1 −9 0 −2

}, { 0 0 0 0 0 0 0 0 0

0 1 2 0 0 1 }, { 0

0 0 0 0 0 0 0 0 0 0

0 0 1 }, { −1 −1 −1 −3 4 −4

2

0 0 −1

1 0 1 }, { 1 1 1 −2 1 1 3 −1 0

0 1 0

0 2 }, { 0 1 0

−2 0 2

−2

2

−1 0

}, { 0 0 0 1 1 0 −2 0 0

2 −2 −1 4 }, { 0 0 0 0 0 0

−1 −1

−1 0 −9 2

}, { 0 0 0 0 0 0 0 0 0

0 −1

−1 −1 0 }, { 0 0 0 0 0 0 0 0 0 0 0 0 −2 0 0 0 }, { 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 }, { −1 −2 −1 0 −1 1 2 −2 −1

1 0

0 0 }, { 1 −1 1

−1 1 0 0 0 −1 0

0 −1 }, { 0 1 0 0 −2 0 0 1 0

1 −2

0 −2 }, { 0 0 0 0 1 0 0 1 −1

0 0 4 −1 0 0 }, { 0 0 0 0 0 0 0 0 0

0 1 0 0

0 }, { 0 0 0 0 0 0 0 0 0 0 0 0 −2 0 0 0 }, { 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 }, { 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }, { 0 0 0

1 −2 0 1 1 0

0 1 0

}, { 0

0 −1

0 1 −1 0 0 1 0 0 0 0 0 }, { 0 0 0 3

0

0 0 0

0 0 0 }, { 0 0 0 0 0 0

0 0

0 2 0 −1 0 1 }, { 0 0 0 0 0 0 0 0 0 0 0 0 −2 1 1

}, { 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }, { 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 }, { 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 },

- If nTrS is equal to 64, stTrSetIdx is equal to 1, and stIdx is equalto 1 the following applies: secTransMatrix [ m ][ n ] = { { −98

7

−1 10 9

1

}, {

7

−100

−7

−4 14

0 2 3 }, { −1 −16

−11

−22 −12 20

−6 −4

}, { 4

4 −2 1 7

−2 24

9

−88 }, { 0

8 −1 −4 0

−1

5

−2

1 7 }, {

2 0 0

2 4

6 −1 −1

1 1

}, { 0 −1

0 −1 0 −1 0

0 0 0

0 4 }, {

1 0 0 1

1 0

0 0

1 0 −4 }, {

89

41 −5

−6 10

16 0 1

}, { −7

−71 −7 9

−46

44 −20

−1 −2 −4 }, {

−15

33

57

24

}, { 0 1

−8

8

−15 −20

−20

}, {

−2

0

0

0 −6 9 −5

}, { 0

−2 −4

0 −3

4 −5 0 0

−2

}, { −1 −1 2 0 1 0 1

1

1 0 −1 4 2 4 }, { 0

0

−1

0 −1

−1 0 0 −2 −1 0 }, {

−24

−9

25

−3 7

}, {

−24

4

−9

11

−7 −18

−66 7 −6 −8 }, { 2

−26

−43

−24

−29 }, {

4

−8 −18

−33

7

47

17 }, { 0 2 2

1

4

20

0

}, { 0 1 −2 −1

−2 2 4

−3 −2 0 1 4 2 }, { 0 1 1 0 0 −1 1

2

4 −1 1 −4 0

}, { 0

−1 0 −1 −1 1 1

−1 0 1 −1 −1 }, {

−11

15

−9 27

7

9

0 }, { −1

17

70

−9 −15 31

}, {

−20 0 −41 2 33

2

−1

}, { 0 −2

44 0 7 −9 44

0

}, { 0

0

4

−2

7 4

}, { 0

−1 0

0 2

0 2

17 }, { 0

0

1 0

1

1

}, { 0

−1 0 1 0

0 0

−2 0 0 1

}, { 0

−1

−5 −5

14 32 −13 3 0 }, { −2

0 9

7 14 16

19 −31

−42 2 }, { 0 1

−8 21

52

0 −6

25 2 −36

}, { −1 −1

−1 18

5

−20 9

0 }, { 0 1 4 2 −9 8

2 −31 −40 1 16

12 }, { 0

1 0 −1 0 −4 9 −4

−9 −10

−17

−2 }, { 0

0 0 −1 1 −3 1 0 0 −3 −5 0

−2 −2 }, { 0

0 0

0

1

0 −2 −2

−3

0 }, { 1 −4

−1

−4 4

2

−14 0

24 −2 }, { 0

1 4 −6 4 7 −3

−4 −30 −4

−11 0

}, { 1 1 −4

5 −1 9

4

−8

5 }, { 0 0 4 2

0 −24

0 2

−2 4

0 }, { 0 0 1 1

3

44

−13

−5 −41

0 }, { 0 0 −1 0

−2 2

1 0

0

}, { 0 0 0 0 0 0 0 2 0 0 2 1 0 −1 −4 0 }, { 0 0 0 0 0 −1 0

0 0

2 0 1 −1 0 }, { 0 −4 0 0

−1 0 −1 4

−2

3 0 }, { −1 1 0 2

7

1 −1 0 }, { 0 0 −1

0 1

−6

9 −12 3 −4

0 }, { 0 0 2

0

21

7

−7

}, { 0 0 0 0 1 0 1

0 0

5 1 16 4 0 }, { 0 0 −1 0

4 −12

0

−2 }, { 0 0 0 0 0 0

1

0 −1 4 0 −4 4 0 }, { 0 0 0 0 0 0 1

0 0 1

0

−2 0 }, { 0

0 −2

0 0

−1 0 0

0 }, { 0 1 0 1 −1 1 2

−2 0 1

−2 0 }, { 0 0

1 −1 0 −4 −4 0

−4 0

−2 0 }, { 0 0 1 0 0 −1 −4 1

6 8

}, { 0 0

0 1 0

0 0 −1

−1 3 0 }, { 0 0 0 0 0 0 2

0 0

0 −6 −1 0 }, { 0 0 0 0 0 0 0 0

0 −1

0 1 3

}, { 0 0 0 0 0 0 0 0 0 0

0 1 −1 0 },

- If nTrS is equal to 64, stTrSetIdx is equal to 1, and stIdx is equalto 2 the following applies: secTransMatrix [ m ][ n ] = { {

−77

−14 20

−2 4 −9

1

}, { −47

−74

−22

7

0 4

}, {

14 12 −22

7

−7 74

−2 −22

}, { −4 1

7 1

−1 1

26

}, { 1

1

5

0 −2 2 1

0 −12 0

}, { −1

−1 2 −1 −2 −1

−2 0 1

0 4 1

}, { 1 1 0

1 1 1

−1

0

0 −2 −2 −4 }, { 0

0 1

−1 −1 0 −1 0 0 −1 0

0 0 }, { −75 −10

44

−1

−1 −2

}, { 41

−1 0 44

−24

−45 −31

−8

}, {

−29 12 −5

−31

−47 }, {

1

−9

20 −1 −2

−12

27 }, { 0

−4 −1

4

3

16 −2

}, { 0 1 1 −2 −2

2 −1

−2 0 −2

−5 0

}, { 0 −1 −2 0

−1 −2

0 −1 1

3 1

}, { 0

0

1 −1 1

−1 0 −2

2 }, {

51

29 −57 44

10 −29

9 −4 }, { −5

11

0

39 −45 −2 −1 43 }, { −11 −4

−39

46 0 10

−35 −5

24 }, {

−16

−12 11 40

}, {

−1

−2

1 −4 4 0 −7

−17

}, {

1 0 2

−2 −1 −4 0

4 0 5 −3

}, {

−1 1 −1 −1 −1 2 0 −2

1 −1 0 −2 1

}, { 0

0 1 1 −1 0 −1 0 0 1 1 0

−1 1 }, {

0

−47

71

−34

}, {

6 50 −49

−30

1

−7 −50 −1

}, {

21 4 51

7

}, { 0 −7

10 −44

1

−14

}, { 0 2 4

−7 1 −2 4 7

21

}, { 0 −1

1 −4 −5 −1 5 2 −4

2 }, { 0 1 1

−1 −1

1

0 −3 −4 2

−2 −4 }, {

0 1 1 0

−1

−1 1

1 0 }, {

7 8 −21

11

14

24

}, {

6

−9

13

−10

−2

6 −36

}, {

−7 1 49

0

−2 −2 −11

4 }, {

−1 9

−8 7

12

−2 −24

1

}, { 0 1 −1 7 7 −7

10 −2

−29

4 }, { 0 −1 1 −4 −1

1 −4

1

4

−2

}, {

0 0 2 1 −1 0 −1 4

0 2

2 0 }, { 0 0 0 −1

1 −1 0 −1 0 1 1 2 1 0 1 }, {

−5

1

−20 1 7 −7 −11 −3

6 −34 9 }, {

−4 0 −9 9

27

29

−14

−17 49

}, { 0 4 11

0

7

27

−6

}, {

4 1 18

−6 −4

−1

}, { 0 −1 −2

−1 7

−2 −25 16

4 43

7 }, { 0

1 0 −1 −3

2 2 −1

2

−7

}, { 0

0

−1 −3

4 1

}, { 0

0 0

−1

0 0 0 −1 0 0 −2 0 }, { 2

1

−2 1

−1 0 10

−1

−17

}, { 1 1 0 4

−6 7

−5 15

}, { 0 −1 −4 9 −4

−4

6 2

}, { 0 −2

1

9 29 −9 −2

60

−1 }, { 0 −1 1

−1 0 −24 4

−8 −5 −8

4 −2 }, { 0 0 0 0 1 0

0 7

−5 0 −2 −18

−1 }, { 0 0 0 −1 0 1

−4

1 −2 4 −2

}, { 0 0 0 0 0 0 0

2

−1 −1

−4 2 0 }, { −1 −1

1 2 −2

7

−4 −3

−3 }, { −1 −1 1 −3 0 2 1 0

−9 1

0

−7

}, { 0 1 1 −2 4 6

−4

7 −8

0 }, { 0 1 1 −1

4 2

−17

0 }, { 0 0 0 0 0

1 −3

2 1 −7

}, { 0 0 0 2 0 0

−1

3 2

19 −4

}, { 0 0 0 0 0 0

1 0

−4 −1 0 }, { 0 0 0 0 0

0

0

1 0 },

- If nTrS is equal to 64, stTrSetIdx is equal to 2, and stIdx is equalto 1 the following applies: secTransMatrix [ m ][ n ] = { {

21

−22 5 5

−9

9

2 }, {

−3

2

0

0

−4

}, { −4 −2 −2

−3 0

−1 −2 66

−2

}, { −5 −1 1

0 0

20 0

−1 0

}, {

0

0 0 0 1

0

1 0 1 −1

}, {

1 −2 0 0 0 −1

0

0 0

}, { 0

0

0 0 0 1 −4 0

0

−1

}, {

0 −1 0 0 0 0 2 0

0 0

}, {

−20

11

−7 −2 −4

}, { −4

−24 −10 −102 0

−24

−17

}, { −2

−2

−1 30 0 −4

0 −6

0 4

}, { 0

0

−1 −8 0

0

17 0

10 22 }, { 0 1

1

5 0 −1

0 −1

0 −1

}, { 0 1 0 0 0 −2 0 1 2 0

0

1 6 }, { 0 1 0 0 0 2 0 0 −2 0

−4 0

2

}, { 0 0 0 0 0 −1 0

0 0

0 0 1

}, {

−6

−1 0

−5 −1 5

0 }, { 5 1

−99

0

−1

−14 4 }, {

−9

10 0

0

0 −6 −77

}, {

0 2

−1 −1 −1 −4

0 0

0 2 11

}, { 0 1

−1 1 1 0

0 −2

0

}, { 0 0

0 0 0 0 −1 0 0

0 0 0

}, { 0 0 1 −1 0 0 0 1 0 0 −1 −1 0

−3 6 }, { 0 0 0 0 0 0 0

0

0 0

−2 }, {

−10 −24 −11 0

8 1 −2

−4 −1 }, {

−5

20 −4

4

1 2 −75

2 }, {

−1

−7 −7 2

0 −14

0 5

}, { 0

0 0 −1 3

1 −6 0 2

1

}, { 0 0

−1

−1 0 0

0 −2

0 1 −4

}, { 0 0 0 0 0 1 0 0

0

−1 0 −2

−6 }, { 0 0 0 0 −1 0 0 0

0

1 0 0 −2

}, { 0 0 0 0 0 0

0 0 0

−1 0 0 −2

}, {

12

6

1

−1 2

1

1 5 0 }, { 2 −1 −4

4

−10

76 4 0

−1 0 }, { 0 −1

−3 −4 −7 −6

1

1 −7 9 −7 }, { 0 0 0 0

1

−4 0 2

0

4 }, { 0 0 0 0 −1 −1 −2

0

0 0

1 −7 }, { 0 0 0 0 0 0 0 0 0 0

−1 0

0 }, { 0 0 0 0 0 0

0 0

0 0 0 2

}, { 0 0 0 0 0 0 0 0 0 0

0 0 0 −1 0 }, { −2

6

1

1

7 1 0 }, {

−1 −1 2

0

−7

1 0 −4 1

}, { 0 0 −1

1 −2 −4 −1

4

2 21 −1 }, { 0 0 0 0 0 0 0 1 −1 0 1 1 1 2 4

}, { 0 0 0 0 0 0

0 −2

−1

2

}, { 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 }, { 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 }, { 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 }, { −2

2 1 −4

1 0

0 −95 0 0 0 }, { 1 0 0

0 0 0

2 4 −15 0 4

}, { 0 0 0 −1 0 0

0 2

1 0

1

0 }, { 0 0 0 0 0 0 0 −1 0 0 −1 0 0

}, { 0

0 0 0 0 0 0 0

0 2

−1

}, { 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

}, { 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 }, { 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }, { −1 1 2 0 1 0 0 −1 0

0 0

−2 0 0 }, { 0

0 1 1 2 0

0 0 1

5 0 0 }, { 0 0 0 0 0 0 0 0 0

0 0

−1

0 }, { 0 0 0 0 0 0 0 0 0 0 −1 0

−1 0

}, { 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 }, { 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }, { 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 }, { 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 },

- If nTrS is equal to 64, stTrSetIdx is equal to 2, and stIdx is equalto 2 the following applies: secTransMatrix [ m ][ n ] = { { −78

−46 −6

−7

0 3

0 −2 −1

}, { 45

−47

7

4

−2

}, {

40

5

−54 17

−16 −4 74 }, { 5 1 −7 −9

−4

9 −4

}, { −1

6 −1 2

−1

0 −4

0 2 }, {

2

−2 −2 1

−1 0

0 0

−1 2 }, { −1 1 2 2 −1 1

0 −1

1 0

5 0 4 }, {

−1

0

0 0

0 0 0 −4 0 0 }, { 75 −36 77 −17 −45

17 1

−2

−1

}, {

32 44 12

42 −6 −44

11 −1 −8

−3 24 }, {

−30

−5

59

42 0 −1 0 −7 21

84 }, {

2 5

4 −2

−6 −1 −5

−2 }, { 0 −2 −4 −1

4

−1 −1 −5 2 0 6

1

}, { −1 1 0 0 1

−4

0 4 −1 0 0 15 0

}, { 0 −1 −2 −1 −1

−1 1 0 1 0

0 4 }, { 0

0 0 1 −1 −1

0

0 0

0 0 }, { −33

0 −49

−20

12

3

4 −2 −3 −2 }, { −7

−38

−7 49 28 1

0

−4 }, { 4 1 12

−54

33 40 −41

24 −21

}, { −1 −1 0 7 −6 7

−4 −18

19 0

7 −14 }, { 1 0

5 −3

1 7 0 −6 0

19

}, { 0

0

−1 1

−3

0 5 1 0

}, { 0

0 −1 2 −1 −1 0 2 1

0 −2

−1 0 }, { 0

0 1

0 0 0 −1

1 0 2

0

}, { 10

−40

−55

−72 −2

−28 −33 2

−4 −3 }, {

7 25

55

2 −15 37

38 −2 2 2

−9 }, {

5

−17 7

13 −36

7 −1 23

}, { 0 −1 −1

1

1 4

−7 0

−9

}, { 0 1 1

−1 −1

4 0

−4 0

4

}, { 0

0 0 −1 1 1 −1 −1 −1 1 0 −7 −11 −1 −2 }, { 0

1 0 0 0 0 1 0

−1 0

4 1 0 }, { 0

0 0 0 0 0

0 0

−2

}, { −4 11 20

33

2 −26 −26 3 −31 −87 −4

0 }, { −2

−5

−34

57 −6 −17 −74 −13 −4 −4

}, { 1

−5

−3 0

1 8

3 −59 −15

−20 }, { 0

1

4

−8

7 9 −2 −1

10

9 }, { 0 −1 −1

−2 0 2 −1 −2 0

1 10 −11

−1 }, { 0

0 0 1

0 2

−1 0

6

1 }, { 0

0 0 −1 0 0

0 1 1 0 1 −1

0 }, { 0

0 0

0 0 0 1 0 0 0 0 0 0 0 }, { 2

9 −19

24 6

0

}, { 1 −2 1 1

2 −5 −17 4 2

4 0

}, { 0

0 7 2

}, { 0

0 0 −2 1 6 1 −2

−1 1 1

}, { 0

0 0 1 1 −1

0 0

2

}, { 0

0 0

0 1 1 0 −2 −1 0

0 0

}, { 0

0 0 0 0 0 −1 0 0

1

−1 −2 0 }, { 0

0 0 0 0 0

0 0

0

0

0 }, { −2

4 −4

2 −21 −10

12

1

}, { 0 −1

−1 4 2

−2 0

12

}, { 0 0 −1 0 −2

−1

0 1 1 −2 −7 −1 0

}, { 0

0

0

0 0 0 1 −1

−1

}, { 0

0 0

−1 0 1

1

−1

−1 1

}, { 0

0

0 0

0 0

0 0 0

0 }, { 0 0

0

0 0

0 0

0

0 −1 0 }, { 0

0

0 0

0 0

0 0 0 1 0 }, { 1 −1

0

2

}, { 0

0 1 −1

0 2 1

1

}, { 0 1

0 0 0 1

0

}, { 0

0

0 1

0 0

0

0 }, { 0

0

0 0

0 0

0 −2 1 1 −1 }, { 0

0

0 0

0 0

0 −2 −1 −1 0 }, { 0 0

0 0 0 0

0 0

0 0 0 1 0 }, { 0 0

0 0 0 0

0

0 0 0

0 },

- If nTrS is equal to 64, stTrSetIdx is equal to 3, and stIdx is equalto 1 the following applies: secTransMatrix [ m ][ n ] = { {

14

−41

10 −14

19

0 0

}, {

19 22

−22

12

−5 0 0 }, {

14

59

−2 4 0

}, {

−2 −16 −24 14 12

12

}, {

2

−1 4 4 1

−4

1

−4 }, { 1 0 −2

0

−4 4 −1 −2

4 1

}, {

1 1

0

2 2

1 0 0 −1 1 −4 }, { 0 0 −1 1 0 −1 0 −1 −2 1

0 −1 1 −2

}, { 40

0

−49

−2

−9

0

}, {

10

18 2

−20 9

−11 −2

}, { −2

−22

15

−1 −69 −26 19

−26

}, { 0 1

4

−24 22 7 6

}, { 0

0

11 4 −9 6 11

}, { 0

4 1

4

0 1

−9 }, { 0

0 0 0 0 −1 1 −2 2 0 −1

}, { 0 0 2 0 1 2

−1

0 0

0

0 −2 }, {

10

−64 −70

0

7 0 }, { 4

14

2

19

−49

}, { −7 7

2

−20 −4 35

−4 6 4

}, { 0

7

−42 −1 7

−46

20

}, { −1

1

16 −9 10

4 5 11

}, { 0

1 0 −7 0 0

−1 −2

−3

−5

}, { 0

−1

1 1 4 −1 1

4 0 }, { 0 −1 0 0

0 0

0 −1 0 0

−1 −6

}, { 1

−14

26

24 4 }, { 0

−10 −47 12 −22

7 −4

}, { 1

−1 −14

−4

42 0 −7

7 21

}, {

−20

17

6

−9

−33 12 1 −3 −2 }, { 0 0 2 −1

−1 −4 −21 1 −17

1

74

}, { 0

−3 1 1 2

4 2

7

1

−1 20 }, { 0 0 0 −1 −2 0 −1 −5

0 0 1 1 2 }, { 0 0 −1 0

1 −1

0

2 5 0

1 2 }, {

}, {

}, {

}, {

}, {

}, { 0 0 −1 0

0

2 −2

−11 1

−5 }, { 0 0 −1 0

1 −4 −4 0 0 −7

4

7 }, { 0 0 0 0

0 0

0 0

−1

1 12 −1 }, {

−1

−4

−2 −12 −2 }, { 0

0 1

7 −2

−2 7

6 2 2 }, { 0 2

0 −2

−9

−1

−24

−31

}, { 0 0

−2

17

1 45

0

}, { 0 0

−1

1 −1 19

−2

}, { 0 0 −1 0

−4 0

−7

0

7

}, { 0 0 0 0 1 0 0

−1 1

−2

−1 25 0 }, { 0

0 0 −1 0

−1 0 −1

1 0 −2

}, { 1

0 1 2 0

−1 0 −2

2 −1 0 }, { 0 1 0 2 −3

−1

−14 −1 −8 0 }, { 0 0 1

−1

−4 0 0

1 −2

}, { 0 0 −1 −1

4 1 −2

−10 −16 2 }, { 0 0 0

1

4 0 0

0 1

}, { 0

1 0

−1

−1 1 0

−2 −10 0 }, { 0

0 0 1 −1

0 1

1

1 8 0 }, { 0 0 0 0

0

−1 0 0 1 0 7 −1

0 }, { 0 −1 −1

−1 1 2 −1

1 −1 −2 −1 −1 0 0 }, { 0 −1

0

0 0 1 −1 0 −2 2 −1 1 −1 0 }, { 0 1 0 0 1 1 −2

0 1 1

0 }, { 0 0 0 0 1

4

10

}, { 0 0 0 0 1 0 0

0 −1 −1 1 −12 −1 −5 0 }, { 0 0 0 0 −1

0 0 −4 1 12 0 0 0 }, { 0 0 0 0 0 0 1 1 0 0 1 1

−1 −1

}, { 0 0 0 0 0 0 0 0 0 0 0 0

0 2 0 },

- If nTrS is equal to 64, stTrSetIdx is equal to 3, and stIdx is equalto 2 the following applies: secTransMatrix [ m ][ n ] = { {

−61 −27

22

7 10 11

12 −2

−1 2 }, {

2 −70

14 −7

−6

−2 −2 }, {

4

−25

10 −2

−4

−47 }, { −4 0 1 −1 3

−3 −3 −2 −1

−1

−5 24 }, { 0

1 4 −5

2

2 2

12

−2 1

}, { −1

0 1 −1 −1 0 −1

−1 1 −2

}, { 0 1 0

1 1 0 0

−1

}, { −1 0 0 0 0 0 0 0 0 0 0 −1 0

−1 2 }, { 47 95 16

−25 −21 9

7 4 −5

}, { 5

40 −14

−67 −19

15

4

}, { −12

−11

16

−11

−44

−69 }, { 1 −1 1 0 −7

−7

−1

1 27 }, { −2 −2 −2 0

1

−4

−1 −5 −4 0 1

}, { 0

0 0

2 1 1

−2

1 4 }, { −1 −1 −1 0 1 1 0 −1 −1 1 0 −2 −2 −1 1

}, { 0 0 0 0 −1 0 1 0 0 −1 −1 2 0 −1 1 2 }, {

7

12

15

1

9 }, {

6 −44

11

−7 14 46

}, {

−11

0 4

−44

−12

−7

−42 }, { 0 0 −2 6 7

0

−9 }, { −1 −2

0

0 −2

1 0 −4

2 −1 2 14 }, { 0 0

2

1 1 −2

−1

0 −2 1 6 −4 }, { 0 −1 2 0

0 −1 0 0 0 −1 1 1 0 1

}, { 0 0 0 1 1 0 0 −1 −2 0

0 −1 0

}, { −21

24

5 91 11

22

2 }, {

7 17

−19 6

−42 }, {

−49 −27

−15

−1

0 }, { −1 −2 6

−1

−21

1

−17

}, { 1 2 0

−2

1 4

−7

14 }, { 0

1

4 −4 1 0 1 −5 −1 0

−1

}, { 0 1 0 −2 −1 0

1 −2 0 1 0 −2

4

}, { 0 0 1 0

1

0 1

0 0 1 0 −1 }, {

−14 #z,899

4

−1 −4

−10 }, {

−7

19

−7

−1

−24

}, { 4 3

−1 11

27

−10

−1

}, { 1

−15

−1

12 −8

−21 }, { 0

−1 0 0 −2 −1

−4

−4

}, { 0 0 0

−2 1 1 −5

0 1 −4

2 4 }, { 0 0

0 0 −1 0 −1 1 −1

−1 1 −1 −2 −3 }, { 0 0 0 −1 −1 0 0 −2 2 1 0 0 −2 −1 1 1 }, {

−2 0

−12 −8 4 1 −2

40

}, { −1

−2

−19 −4

}, { 1 0

−1

0 0

}, { 1 1

14 −14

−4 −19

−1 }, { 0 0

−1 2 −2 −2 9

−10

}, { 0 0

0 0

1 1

7

−1

}, { 0 0 0 0 0 1 0 0

−1

0 −6

−1

}, { 0 0 0 0 0 −2 0 1 0

1 0 0 −1 −1 0 }, { z,899; 0

0 4 −1

0 0 −19 1

}, { 0 −1

−1

0

−1

−1 −4

}, { 0

0 −1

−17

2

0

}, { 0 0

1 −7

7 −2

0

0

}, { 0

0 0 1

1

−4

−2

4 0

}, { 0 0 0 0 0

0 1 0 0

0 0 −1

}, { 0 0 0

0

0 0 −1

0 −1 1

}, { 0 0 0 0 0 0 0 1 0 0

0 0 0

0 }, {

−1

0 0 −2

0

−4 1 −12 5

}, { 0

0 −1

0 1

−6 0 9 −19 1

}, { 1

0

−2 7 −1

0 −40 −4 0

}, { 0

0 0 0

0 −1 −1

1 1 −1 7 1 −1 }, { 0 0 0 −1 0

1 0 1

5 0

−1 −1 }, { 0

0 0

0 0 −1 0

1 0 −2

1 −1 }, { 0 0 0 0 0 0 0 0 0 1 1 0

0 0 0 }, { 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 −1 },

indicates data missing or illegible when filed

TABLE 9-4 Syntax elements and associated binarizations BinarizationSyntax element Process Input parameters Syntax . . . . . . . . .structure st_idx[ ][ ] TR cMax = 2, cRiceParam = 0

TABLE 9-10 Assignment of ctxInc to syntax elements with context codedbins

Syntax element 0 1 2 3

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

indicates data missing or illegible when filed

9.2.4.2.8 Derivation Process of ctxInc for the Syntax Element st_idx

Inputs to this process are the colour component index cIdx, the luma orchroma location (x0, y0) specifying the top-left sample of the currentluma or chroma coding block relative to the top-left sample of thecurrent picture depending on cIdx, the tree type treeType, the lumaintra prediction mode IntraPredModeY[x0] [y0] as specified in clause8.2.2, the syntax element Intra_chroma_pred_mode[x0] [y0] specifying theintra prediction mode for chroma samples as specified in clause 7.4.5.6,and the multiple transform selection flag tu_mts_flag[x0] [y0].

Output of this process is the variable ctxInc.

The variable intraModeCtx is derived as follows:

If cIdx is equal to 0, intraModeCtx is derived as follows:

-   -   intraModeCtx={IntraPredModeY[x0] [y0]<=1}!1:0

The variable mtsCtx is derived as follows:

-   -   mtsCtk=(tu_mts_flag[x0] [y0]==1&& treeType 1=SINGLE_TREE)!1:0

The variable ctxInc is derived as follows:

-   -   ctxInc=(bixIdx<<1)+intraModeCtx+(mtsCtx<<2)

A transform kernel matrix of Table 44 is a transform kernel matrixapplied to an inverse secondary transform performed in a decodingapparatus, rather than a kernel matrix for a forward secondarytransform. Therefore, an encoding apparatus may perform a secondarytransform based on the inverse process of the transform process shown inTable 44, in which case the encoding apparatus may perform an RST usingthe transposed matrix of the transform kernel matrix of Table 44.

FIG. 10 is a flowchart illustrating an operation of a video decodingapparatus according to an embodiment of the present disclosure.

Each operation illustrated in FIG. 10 may be performed by the decodingapparatus 300 illustrated in FIG. 3. Specifically, S1010 may beperformed by the entropy decoder 310 illustrated in FIG. 3, S1020 may beperformed by the dequantizer 321 illustrated in FIGS. 3, S1030 and S1040may be performed by the inverse transformer 322 illustrated in FIGS. 3,and S1050 may be performed by the adder 340 illustrated in FIG. 3.Operations according to S1010 to S1050 are based on some of theforegoing details explained with reference to FIG. 4 to FIG. 9.Therefore, a description of specific details overlapping with thoseexplained above with reference to FIG. 3 to FIG. 9 will be omitted orwill be made briefly.

The decoding apparatus 300 according to an embodiment may derivequantized transform coefficients for a target block from a bitstream(S1010). Specifically, the decoding apparatus 300 may decode informationon the quantized transform coefficients for the target block from thebitstream and may derive the quantized transform coefficients for thetarget block based on the information on the quantized transformcoefficients for the target block. The information on the quantizedtransform coefficients for the target block may be included in asequence parameter set (SPS) or a slice header and may include at leastone of information on whether a reduced transform (RST) is applied,information on a reduced factor, information on a minimum transform sizeto which the RST is applied, information on a maximum transform size towhich the RST is applied, information on a reduced inverse transformsize, and information on a transform index indicating any one oftransform kernel matrices included in a transform set.

The decoding apparatus 300 according to an embodiment may derivetransform coefficients by dequantizing the quantized transformcoefficients for the target block (S1020).

The derived transform coefficients may be arranged in 4×4 blocksaccording to an inverse diagonal scanning order, and transformcoefficients in a 4×4 block may also be arranged according to theinverse diagonal scanning order. That is, the dequantized transformcoefficients may be disposed according to the inverse scanning orderapplied in video codec, such as VVC or HEVC.

The decoding apparatus 300 according to an embodiment may derivemodified transform coefficients based on an inverse reduced secondarytransform (RST) of the transform coefficients (S1030).

In an example, the inverse RST may be performed based on an inverse RSTtransform matrix, and the inverse RST transform matrix may be anonsquare matrix in which the number of columns is less than the numberof rows.

In an embodiment, S1030 may include decoding a transform index,determining whether a condition for applying an inverse RST is satisfiedbased on the transform index, selecting a transform kernel matrix, andapplying the inverse RST to the transform coefficients based on theselected transform kernel matrix and/or the reduced factor when thecondition for applying the inverse RST is satisfied. In this case, thesize of an inverse RST matrix may be determined based on the reducedfactor.

The decoding apparatus 300 according to an embodiment may deriveresidual samples for the target block based on an inverse transform ofthe modified transform coefficients (S1040).

The decoding apparatus 300 may perform an inverse primary transform onthe modified transform coefficients for the target block, in which casea reduced inverse transform may be applied or a conventional separabletransform may be used as the inverse primary transform.

The decoding apparatus 300 according to an embodiment may generatereconstructed samples based on the residual samples for the target blockand prediction samples for the target block (S1050).

Referring to S1030, it may be identified that the residual samples forthe target block are derived based on the inverse RST of the transformcoefficients for the target block. From the perspective of the size ofthe inverse transform matrix, since the size of a regular inversetransform matrix is N×N but the size of the inverse RST matrix isreduced to N×R, it is possible to reduce memory usage in a case ofperforming the inverse RST by an R/N ratio compared to that in a case ofperforming a regular transform. Further, using the inverse RST matrixcan reduce the number of multiplications (N×R) by the R/N ratio,compared to the number of multiplications N×N in a case of using theregular inverse transform matrix. In addition, since only R transformcoefficients need to be decoded when the inverse RST is applied, thetotal number of transform coefficients for the target block may bereduced from N to R, compared to that in a case where N transformcoefficients needs to be decoded when a regular inverse transform isapplied, thus increasing decoding efficiency. That is, according toS1030, the (inverse) transform efficiency and decoding efficiency of thedecoding apparatus 300 may be increased through the inverse RST.

FIG. 11 is a control flowchart illustrating an image decoding method bya decoding apparatus according to an embodiment of the presentdisclosure. A method for transforming an image, specifically a secondarytransform process, performed by the decoding apparatus or an inversesecondary transform corresponding to a secondary transform performed byan encoding apparatus is described with reference to FIG. 11.Hereinafter, an inverse secondary transform performed by the decodingapparatus is referred to as a non-separable secondary transform.

The decoding apparatus 300 receives information on quantized transformcoefficients, an intra prediction mode, and a transform index for anon-separable secondary transform through a bitstream (S1100).

According to an embodiment, the non-separable secondary transform is anon-separable transform of transforming coefficients without separatingthe coefficients in a specific direction, unlike a primary transform ofvertically or horizontally separating coefficients to be transformed andtransforming the coefficients. The non-separable transform may be alow-frequency non-separable transform of transforming only alow-frequency region rather than an entire target block to betransformed.

The decoding apparatus 300 further receives flag information indicatingwhether a transform index exists through the bitstream.

The flag information indicating whether the transform index is receivedmay be sps_st_enabled_flag in Table 41, which may be modified tosps_1fnst_enabled flag according to the type of a secondary transform.The flag information may indicate whether the transform index isreceived, that is, whether the transform index exists in the bitstream,and may be received via the syntax of a sequence parameter.

When the flag information is 0, no transform index exists, and thus thenon-separable secondary transform may not be performed. When the flaginformation is 1, the transform index exists, and thus the transformindex may be received and parsed by the decoding apparatus.

The transform index may exist in the syntax of a coding unit.

A syntax element of the transform index according to an embodiment mayindicate whether the non-separable secondary transform is applied andany one of transform kernel matrices included in a transform set. Whenthe transform set includes two transform kernel matrices, the syntaxelement of the conversion index may have three values.

That is, according to an embodiment, the value of the syntax element ofthe transform index may include 0 indicating that the non-separablesecondary transform is not applied to the target block, 1 indicating afirst transform kernel matrix of the transform kernel matrices, and 2indicating a second transform kernel matrix of the transform kernelmatrices. This information is received as syntax information, and thesyntax information is received as a binarized bin string including a 0and a 1.

In this case, the three values of the syntax element of the transformindex may be coded into 0, 10, and 11, respectively, according totruncated unary coding. That is, the value of the syntax element of 0may be binarized into ‘0’, the value of the syntax element of 1 may bebinarized into ‘10’, and the value of the syntax element of 2 may bebinarized into ‘11’.

According to an embodiment, different pieces of context information,that is, different probability models, may be applied to two bins of thetransform index, respectively. That is, all of the two bins of thetransform index may be decoded by a context method rather than by abypass method, wherein a first bin of the bins of the syntax element ofthe transform index may be decoded based on first context information,and a second bin of the bins of the syntax element of the transformindex may be decoded based on second context information.

Transform coefficients may be derived from the quantized transformcoefficients received through the bitstream via dequantization as shownin S1020 of FIG. 10 (S1110). The following transform coefficients referto the dequantized transform coefficients.

When the received transform index does not indicate that thenon-separable secondary transform is not performed, that is, when thetransform index is not 0, the decoding apparatus may derive an inputtransform coefficient size indicating the length of the dequantizedtransform coefficient to which the non-separable secondary transform isapplied, an output transform coefficient size indicating the length of amodified transform coefficient to which the non-separable secondarytransform has been applied, and a transform set mapped to an intra modefor the target block (S1120).

As illustrated in 8.5.4.4. of Table 44, the input transform coefficientsize may be represented by ‘nonZeroSize’, and the output transformcoefficient size may be represented by ‘nTrS’.

The size of an input transform coefficient refers to the length oftransform coefficients, that is, the number of transform coefficients,which are subjected to an matrix operation with a transform kernelmatrix, and the size of an output transform coefficient refers to thelength of modified transform coefficients, that is, the number ofmodified transform coefficients, output after the matrix operation isperformed.

According to an example, when the size of the target block is 4×4 or8×8, the size of the input transform coefficient may be 8, and when thesize of the target block is not 4×4 or 8×8, the size of the inputtransform coefficient may be 16. That is, when the size of the targetblock, that is, the transform block, is 4×4, eight transformcoefficients arranged in a scanning order from a top-left position ofthe 4×4 block are input data, and when the size of the transform blockis 8×8, only eight transform coefficients arranged in the scanning orderfrom a top-left position of the 8×8 block are input data. In cases otherthan these two cases, that is, 1) when the target block is 4×N, N×4(N≥8), or 2) when the target block has both a width and a height equalto or greater than 8 (while 8 or greater) and the width or the height isgreater than 8, 16 transform coefficients are input for a matrixoperation.

According to an example, when the width and height of the target blockare 8 or greater, the size of the output transform coefficient may be48, and when the width or height of the target block is less than 8, thesize of the output transform coefficient may be 16.

For example, when the width and height of the transform block are 8 orgreater, inverse RST 8×8 is applied. That is, the non-separablesecondary transform may be applied to up to a top-left 4×4 region of atop-left 8×8 region of the transform block, and 48 modified transformcoefficients may be derived in top-left, top-right, and bottom-left 4×4regions excluding a bottom-right 4×4 region rather than in the entire8×8 region as a result of the non-separable secondary transform.However, when the width or height of the target block is less than 8,for example, in a case of a 4×4, 4×8, or 8×4 transform block, inverseRST 4×4 is applied to a top-left 4×4 region of the transform block. Thatis, the non-separable secondary transform may be applied to 8 or 16transform coefficients arranged according to the scanning order from atop-left position of the 4×4 region, and 16 modified transformcoefficients may be derived in the 4×4 region as a result of thenon-separable secondary transform.

The transform set may be derived by a mapping relationship according tothe intra prediction mode for the target block, and one transform setmay be mapped to a plurality of intra prediction modes. For example, asillustrated in 8.5.4.5 of Table 44, there may be four transform setsaccording to the intra prediction mode.

When the input data for the non-separable secondary transform isderived, the decoding apparatus may derive a transform kernel matrixbased on the output transform coefficient size, the transform set, andthe transform index (S1130).

Each one transform set may include a plurality of transform kernelmatrices. The transform index may indicate any one of the plurality oftransform kernel matrices. For example, when one transform set includestwo transform kernel matrices, the transform index may indicate any oneof the two transform kernel matrices.

The transform kernel matrix may be determined based on the number ofmodified transform coefficients, information on the transform set, andthe value of the transform index. Table 44 shows that a transform matrixis determined based on the transform output length nTrS, that is, thenumber of modified transform coefficients output through an matrixoperation with a transform kernel matrix, information stTrSetldx on atransform set mapped to an intra prediction mode stIntraPredMode for atarget block, and a transform index value stldx.

As shown in Table 8, Table 9, Table 14, Table 18, and Table 44, the sizeof the transform kernel matrix and the selected matrix itself may changedepending on the type of the non-separable secondary transform (RST 8×8or RST 4×4) applied to a block having a predetermined size in the targetblock and the number of output modified transform coefficients.

According to an example, the transform kernel matrix may be applied to aspecific region, for example, an 8×8 region or a 4×4 region, at thetop-left of the target block according to the reduced or simplified sizeof the secondary transform, and the size of the modified transformcoefficients, that is, the number of the transform coefficients, outputby applying the transform kernel matrix may be derived based on thetransform index, the intra prediction mode, and the size of the targetblock to which the non-separable secondary transform is applied.

According to an example, when the non-separable secondary transform isapplied to transform coefficients in a region, that is, an 8×8 region ora 4×4 region, of the target block, the non-separable secondary transformmay be applied to only some of the transform coefficients included inthe 8×8 region or the 4×4 region. When only 48 transform coefficientsamong the transform coefficients in the 8×8 region are output for thenon-separable secondary transform, a 64 x m transform kernel matrixapplied to the 8×8 region may be further reduced to a 48 x m transformkernel matrix. Alternatively, when only eight transform coefficientsamong the transform coefficients in the 4×4 region are input for thenon-separable secondary transform, a transform kernel matrix applied tothe 4×4 region is a 16×8 matrix.

According to an example, m may be 16, and a 48×16 transform kernelmatrix may be a transform kernel matrix based on Table 14, that is, thetransposed matrix of the matrix of Table 14. Alternatively, according toan example, a 16×8 transform kernel matrix may be a transform kernelmatrix based on Table 18. A 16×8 transform kernel matrix including onlyeight columns from the left in a 16×16 matrix obtained by transposingthe matrix of Table 18 may be applied. Alternatively, a 48×8 matrixincluding only eight columns from the left in a 48×16 matrix obtained bytransposing the matrix of Table 14 may be applied.

In summary, when the size of the input transform coefficients is 8 andthe size of the output transform coefficients is 16, a matrix includingeight columns extracted from a preset 16×16 transform kernel matrix maybe used for the matrix operation. When the size of the input transformcoefficient is 16 and the size of the output transform coefficient is16, the preset 16×16 transform kernel matrix may be used for the matrixoperation. When the size of the input transform coefficient is 16 andthe size of the output transform coefficient is 48, a preset 48×16transform kernel matrix may be used for the matrix operation. When thesize of the input transform coefficient is 8 and the size of the outputtransform coefficient is 48, a matrix including eight columns extractedfrom the preset 48×16 transform kernel matrix may be used for the matrixoperation.

There are four transform sets and each transform set may include twotransform kernel matrices. In this case, the transform index may be 0indicating that no secondary transformation is applied, 1 or 2indicating any one of the transform kernel matrices.

The decoding apparatus may derive the modified transform coefficientsbased on the matrix operation of the transform kernel matrix and atransform coefficient list corresponding to the input transformcoefficient size (S1140).

The transform coefficient list may include dequantized transformcoefficients that are read according to the forward diagonal scanningorder of the target block.

The modified transform coefficients in a two-dimensional array may bederived through the matrix operation of a one-dimensional array of thetransform coefficients derived through the dequantization, that is, thetransform coefficient list, and the transform kernel matrix as shown inEquation 7.

According to this embodiment, the inverse transformer 322 may apply thetransform kernel matrix to transform coefficients in the top-left 4×4region of the 8×8 region of the target block, thereby deriving themodified transform coefficients in the top-left 4×4 region, thetop-right 4×4 region, and the bottom-left 4×4 region of the 8×8 region.

According to an example, when performing the matrix operation of thetransform coefficients in the top-left 4×4 region of the 8×8 region andthe transform kernel matrix, the transform coefficients in the top-left4×4 region of the 8×8 region may be one-dimensionally arranged accordingto the forward diagonal scanning order as shown in Table 16, and afterthe matrix operation with the transform kernel matrix, the transformcoefficients in the one-dimensional array may be two-dimensionallyarranged in the top-left 4×4 region, the top-right 4×4 region, and thebottom-left 4×4 region of the 8×8 region according to either therow-first order or the column-first order according to the intraprediction mode applied to the target block as shown in Table 15 orTable 17. That is, an inverse secondary transform may be applied to 16transform coefficients in the top-left 4×4 region of the 8×8 region, and48 modified transform coefficients in the top-left 4×4 region, thetop-right 4×4 region, and the bottom-left 4×4 region of the 8×8 regionmay be derived through the operation with the transform kernel matrix.

According to an embodiment, the inverse transformer 322 may apply thetransform kernel matrix to some transform coefficients in a 4×4 regionto which a forward LFNST is applied in the target block, for example, upto eight transform coefficients from the top-left position of the 4×4region according to the scanning order, thereby deriving 16 modifiedtransform coefficients in the 4×4 region. Hereinafter, a region in whichthe eight transform coefficients are arranged is referred to as atop-left region in the 4×4 region.

As described above, when either the height or the width of the targetblock to which the transform is applied is less than 8, a non-separablesecondary transform with a transform matrix having a reduced size may beapplied, for example, to a 4×4 transform block, an upper 4×4 block of a4×8 transform block, or a left 4×4 block of a 8×4 transform block.

According to an example, when performing the matrix operation of thetransform coefficients in the top-left region of the 4×4 region and thetransform kernel matrix, the eight transform coefficients in thetop-left region of the 4×4 region may be one-dimensionally arrangedaccording to the forward diagonal scanning order, and after the matrixoperation with the transform kernel matrix, the transform coefficientsin the one-dimensional array may be two-dimensionally arranged in the4×4 region according to either the row-first order or the column-firstorder according to the intra prediction mode applied to the target blockas shown in Table 12 or Table 13. That is, an inverse secondarytransform may be applied to the eight transform coefficients in the 4×4region, and 16 modified transform coefficients in the 4×4 region may bederived through the operation with the transform kernel matrix.

When an intra prediction mode applicable to the target block is any oneof 65 directional modes, intra prediction modes are symmetric withrespect to intra prediction mode 34 in a top-left diagonal direction,and the intra prediction mode applied to the target block includes modes2 to 34 in a left direction based on intra prediction mode 34, themodified transform coefficients may be two-dimensionally arrangedaccording to the row-first order.

When the intra prediction mode applied to the target block includesmodes 35 to 66 in a right direction based on intra prediction mode 34,the modified transform coefficients may be two-dimensionally arrangedaccording to the column-first order.

When the intra prediction mode applied to the target block is a planarmode or a DC mode, the modified transform coefficients may betwo-dimensionally arranged according to the row-first order.

The inverse transformer 322 may apply the non-separable secondarytransform, thereby generating the modified transform coefficients in the8×8 region, specifically the 8 x 8 region excluding the bottom-right 4×4region of the 8×8 region, or in the 4×4 region as a two-dimensionalblock.

When the modified transform coefficients are derived as a result of thenon-separated secondary transform, the modified transform coefficientsmay be clipped to values within a predetermined range (S1150).

According to an example, the modified transform coefficients may beclipped based on Equation 9, and a maximum value (maxInvSecTr) and aminimum value (minInvSecTr) indicating a clipping range may be set asshown in Equation 13.

At this time, the modified transform coefficients may be scaled andclipped according to Equation 10, and according to an example, S may be7, and in this case, the modified transform coefficients may beconsidered to be scaled by 128.

The decoding apparatus may derive residual samples for the target blockbased on the inverse primary transform with respect to the clippedmodified transform coefficients (S1160).

According to an embodiment of the present document, the inverse primarytransform may be based on multiple transform selection (MTS). Amulti-core transform to which multiple transforms are applied as aprimary transform refers to a method of transforming using DCT (DiscreteCosine Transform) type 2 and DST (Discrete Sine Transform) type 7, DCTtype 8, and/or DST type 1 additionally. The modified transformcoefficients in the frequency domain according to this inverse primarytransform are transformed into residual signals in the spatial domain.

The decoding apparatus may clip the residual samples derived based onthe inverse primary transform to values within a predetermined range(S1170).

A clipping range applied to the modified transform coefficient and aclipping range applied to the residual sample may be set to the samerange. Alternatively, according to another example, the clipping rangemay be set differently according to the inverse secondary transform andthe inverse primary transform.

Also, according to another embodiment of the present document, clippingof the residual samples may not be performed.

FIG. 12 is a flowchart illustrating an operation of a video encodingapparatus according to an embodiment of the present disclosure.

Each operation illustrated in FIG. 12 may be performed by the encodingapparatus 200 illustrated in FIG. 2. Specifically, S1210 may beperformed by the predictor illustrated in FIG. 2, S1220 may be performedby the subtractor 231 illustrated in FIGS. 2, S1230 and S1240 may beperformed by the transformer 232 illustrated in FIGS. 2, and S1250 maybe performed by the quantizer 233 and the entropy encoder 240illustrated in FIG. 2. Operations according to S1210 to S1250 are basedon some of contents described in FIG. 4 to FIG. 9. Therefore, adescription of specific details overlapping with those explained abovewith reference to FIG. 2 and FIG. 4 to FIG. 9 will be omitted or will bemade briefly.

The encoding apparatus 200 according to an embodiment may deriveprediction samples based on an intra prediction mode applied to a targetblock (S1210).

The encoding apparatus 200 according to an embodiment may deriveresidual samples for the target block (S1220).

The encoding apparatus 200 according to an embodiment may derivetransform coefficients for the target block based on primary transformof the residual samples (S1230). The primary transform may be performedthrough a plurality of transform kernels, and the transform kernels maybe selected based on the intra prediction mode.

The decoding apparatus 300 may perform a secondary transform,specifically an NSST, on the transform coefficients for the targetblock, in which case the NSST may be performed based on a reducedtransform (RST) or without being based on the RST. When the NSST isperformed based on the reduced transform, an operation according toS1240 may be performed.

The encoding apparatus 200 according to an embodiment may derivemodified transform coefficients for the target block based on the RST ofthe transform coefficients (S1240). In an example, the RST may beperformed based on a reduced transform matrix or a transform kernelmatrix, and the reduced transform matrix may be a nonsquare matrix inwhich the number of rows is less than the number of columns.

In an embodiment, S1240 may include determining whether a condition forapplying the RST is satisfied, generating and encoding the transformindex based on the determination, selecting a transform kernel, andapplying the RST to the residual samples based on the selected transformkernel matrix and/or a reduced factor when the condition for applyingthe RST is satisfied. In this case, the size of the reduced transformkernel matrix may be determined based on the reduced factor.

The encoding apparatus 200 according to an embodiment may derivequantized transform coefficients by performing quantization based on themodified transform coefficients for the target block and may encodeinformation on the quantized transform coefficients (S1250).

Specifically, the encoding apparatus 200 may generate the information onthe quantized transform coefficients and may encode the generatedinformation on the quantized transform coefficients.

In an example, the information on the quantized transform coefficientsmay include at least one of information on whether the RST is applied,information on the reduced factor, information on a minimum transformsize to which the RST is applied, and information on a maximum transformsize to which the RST is applied.

Referring to S1240, it may be identified that the transform coefficientsfor the target block are derived based on the RST of the residualsamples. From the perspective of the size of the transform kernelmatrix, since the size of a regular transform kernel matrix is N×N butthe size of the reduced transform matrix is reduced to R×N, it ispossible to reduce memory usage in a case of performing the RST by anR/N ratio compared to that in a case of performing a regular transform.Further, using the reduced transform kernel matrix can reduce the numberof multiplications (R×N) by the R/N ratio, compared to the number ofmultiplications N×N in a case of using the regular transform kernelmatrix. In addition, since only R transform coefficients are derivedwhen the RST is applied, the total number of transform coefficients forthe target block may be reduced from N to R, compared to that in a casewhere N transform coefficients are derived when a regular transform isapplied, thus reducing the amount of data transmitted by the encodingapparatus 200 to the decoding apparatus 300. That is, according toS1240, the transform efficiency and coding efficiency of the encodingapparatus 320 may be increased through the RST.

FIG. 13 is a control flowchart illustrating an image encoding method byan encoding apparatus according to an embodiment of the presentdisclosure. A method for transforming an image, specifically a secondarytransform process, performed by the encoding apparatus or a secondarytransform corresponding to an inverse secondary transform performed by adecoding apparatus is described with reference to FIG. 13. Hereinafter,a secondary transform performed by the encoding apparatus is referred toas a non-separable secondary transform.

According to an embodiment, the non-separable secondary transform is anon-separable transform of transforming coefficients without separatingthe coefficients in a specific direction, unlike a primary transform ofvertically or horizontally separating coefficients to be transformed andtransforming the coefficients. The non-separable transform may be alow-frequency non-separable transform (LFNST) of transforming only alow-frequency region rather than an entire target block to betransformed.

First, the encoding apparatus 200 derives transform coefficients byapplying a primary transform to residual samples for a target block(S1300).

The encoding apparatus 200 may clip the derived transform coefficientsto values within a predetermined range (S1310).

A clipping process applied to the primary transform coefficient may beomitted according to an embodiment.

When a non-separable secondary transform is applied to the transformcoefficients derived through the primary transform, the encodingapparatus derives the size of an input transform coefficient, the sizeof an output transform coefficient, and a transform set mapped to anintra mode for the target block (S1320).

A transform process performed by the encoding apparatus is the reverseof a transform process performed by the decoding apparatus. Thus,referring to 8.5.4.4. of Table 44, the size of the input transformcoefficient may be represented by ‘nTrS’, and the size of the outputtransform coefficient may be represented by ‘nonZeroSize’.

The size of the input transform coefficient refers to the length oftransform coefficients, that is, the number of transform coefficients,which are subjected to an matrix operation with a transform kernelmatrix, and the size of the output transform coefficient refers to thelength of modified transform coefficients, that is, the number ofmodified transform coefficients, output after the matrix operation isperformed.

According to an example, when the width and height of the target blockare 8 or greater, the size of the input transform coefficient may be 48,and when the width or height of the target block is less than 8, thesize of the input transform coefficient may be 16.

For example, when the width and height of the transform block are 8 orgreater, RST 8×8 may be applied. Thus, the non-separable secondarytransform may be applied to a top-left 8×8 region of the transformblock, and 8 (e.g., an 8×8 transform block) or 16 (e.g., a transformblock greater than 8×8) modified transform coefficients may be derivedas a result of the non-separable secondary transform. However, when thewidth or height of the target block is less than 8, for example, in acase of a 4×4, 4×8, or 8×4 transform block, RST 4 x 4 may be applied.Thus, the non-separable secondary transform may be applied to 16transform coefficients in a top-left 4×4 region of the transform block,and 8 (e.g., a 4×4 transform block) or 16 (e.g., a 4×8 or 8×4 transformblock) modified transform coefficients may be derived as a result.

According to an example, when the size of the target block is 4×4 or8×8, the size of the output transform coefficient may be 8, and when thesize of the target block is not 4×4 or 8×8, the size of the outputtransform coefficient may be 16. That is, when the size of the targetblock, that is, the transform block, is 4×4, eight pieces of data areoutput after the non-separable secondary transform, and even when thesize of the transform block is 8×8, only eight transform coefficientsare output after the non-separable secondary transform. In cases otherthan these two cases, that is, 1) when both the width and the height areequal to or greater than 8 and at least one of the width the height isgreater than 8, or 2) when the target block is 4×N or N×4 (N≥8), 16transform coefficients may output in each matrix operation.

The transform set may be derived by a mapping relationship according tothe intra prediction mode for the target block, and one transform setmay be mapped to a plurality of intra prediction modes. For example,there may be four transform sets according to the intra prediction modeas illustrated in 8.5.4.5 of Table 44.

When the input data for the non-separable secondary transform isderived, the encoding apparatus may derive the modified transformcoefficients based on a matrix operation of any one of transform kernelmatrices included in the transform set and a transform coefficientcorresponding to the size of the input transform coefficient (S1330).

The transformer 232 of the encoding apparatus may select any one of aplurality of transform kernel matrices included in the transform set.

According to an embodiment, the transform set may be derived by themapping relationship according to the intra prediction mode for thetarget block, and one transform set may be mapped to a plurality ofintra prediction modes. Further, each one transform set may include aplurality of transform kernel matrices. When one transform set includestwo transform kernel matrices, a transform index indicating any one ofthe two transform kernel matrices may be encoded and signaled to adecoding apparatus.

When two transform processes are applied to the residual samples, theresidual samples may be referred to as transform coefficients after theprimary transform, and may be referred to as modified transformcoefficients after the primary transform and then the non-separablesecondary transform.

Each one transform set may include a plurality of transform kernelmatrices. The transform index may indicate any one of the plurality oftransform kernel matrices. For example, when one transform set includestwo transform kernel matrices, the transform index may indicate any oneof the two transform kernel matrices.

The transform kernel matrix may be determined based on the number ofmodified transform coefficients, information on the transform set, andthe value of the transform index. Table 44 shows that a transform matrixis determined based on the transform output length nTrS, that is, thenumber of modified transform coefficients output through an matrixoperation with a transform kernel matrix, information stTrSetldx on atransform set mapped to an intra prediction mode stIntraPredMode for atarget block, and a transform index value stldx.

As shown in Table 8, Table 9, Table 14, Table 18, and Table 44, the sizeof the transform kernel matrix and a matrix coefficient may changedepending on the type of the non-separable secondary transform (RST 8×8or RST 4×4) applied to a block having a predetermined size in the targetblock and the number of output modified transform coefficients.

According to an example, the transform kernel matrix may be applied to aspecific region, for example, an 8×8 region, specifically the 8×8 regionexcluding a bottom-right 4×4 region of the 8×8 region, or a 4×4 region,at the top-left of the target block according to the reduced orsimplified size of the secondary transform, and the size of the modifiedtransform coefficients, that is, the number of the transformcoefficients, output by applying the transform kernel matrix may bederived based on the transform index, the intra prediction mode, and thesize of the target block to which the non-separable secondary transformis applied.

According to an example, when the non-separable secondary transform isapplied to transform coefficients in a region, that is, an 8×8 region ora 4×4 region, of the target block, the non-separable secondary transformmay be applied to only some of the transform coefficients included inthe 8×8 region or the 4×4 region. When only 48 transform coefficientsamong the transform coefficients in the 8×8 region are input for thesecondary transform, an m×64 transform kernel matrix applied to the 8×8region may be further reduced to an m×48 transform kernel matrix.Alternatively, when only eight transform coefficients among thetransform coefficients in the 4×4 region are output by applying thenon-separable secondary transform, a transform kernel matrix applied tothe 4×4 region is an 8×16 matrix.

According to an example, m may be 16, and a 16×48 transform kernelmatrix may be a transform kernel matrix illustrated in Table 14.Alternatively, according to an example, an 8×16 transform kernel matrixmay be a transform kernel matrix based on Table 18. That is, when mtransform coefficients are generated by applying the secondary transformto a 4×4 region, an m×16 transform kernel matrix may be applied to the4×4 region. According to an example, m may be 8, and an 8×16 transformkernel matrix may be a matrix including top eight rows in Table 18.Alternatively, according to an example, an 8×48 transform kernel matrixmay be a transform kernel matrix based on Table 14. That is, when mtransform coefficients are generated by applying the secondary transformto an 8×8 region excluding the bottom-right 4×4 region, an m x 48transform kernel matrix may be applied to the 8×8 region excluding thebottom-right 4×4 region. According to an example, m may be 8, and an8×48 transform kernel matrix may be a matrix including top eight rows inTable 14.

In summary, when the size of the input transform coefficients is 16 andthe size of the output transform coefficients is 8, a matrix includingeight rows extracted from a preset 16×16 transform kernel matrix may beused for the matrix operation. When the size of the input transformcoefficient is 16 and the size of the output transform coefficient is16, the preset 16×16 transform kernel matrix may be used for the matrixoperation. When the size of the input transform coefficient is 48 andthe size of the output transform coefficient is 16, a preset 16×48transform kernel matrix may be used for the matrix operation. When thesize of the input transform coefficient is 48 and the size of the outputtransform coefficient is 8, a matrix including eight rows extracted fromthe preset 16×48 transform kernel matrix may be used for the matrixoperation.

There are four transform sets and each transform set may include twotransform kernel matrices. In this case, the transform index may be 0indicating that no secondary transformation is applied, 1 or 2indicating any one of the transform kernel matrices.

When performing the non-separable secondary transform on the transformcoefficients using the transform kernel matrix, the transformer 232 mayone-dimensionally arrange the transform coefficients in atwo-dimensional array via the primary transform according to either arow-first order or a column-first order based on the intra predictionmode applied to the target block.

Specifically, according to an embodiment, the transformer 232 may applythe transform kernel matrix to transform coefficients in the top-left4×4 region, the top-right 4×4 region, and the bottom-left 4×4 region ofthe 8×8 region of the target block, thereby deriving modified transformcoefficients corresponding to the top-left 4×4 region of the 8×8 region.

The transform kernel matrix may be applied to a specific region at thetop-left of the target block, for example, an 8×8 region, a 4×4 region,or a portion of the 8×8 region, depending on the reduced or simplifiedsize of the secondary transform, and the size of modified transformcoefficients, that is, the number of modified transform coefficients,output by applying the transform kernel matrix may be derived based onthe size of the transform kernel matrix, the intra prediction mode, andthe size of the target block to which the secondary transform isapplied.

As shown in Equation 5, the two-dimensional transform coefficients needto be arranged in one dimension for the matrix operation with thetransform kernel matrix, and a smaller number of modified transformcoefficients than the number of transform coefficients may be derivedthrough the operation illustrated in Equation 6.

That is, the transform coefficients in a two-dimensional array in thespecific region may be read in one dimension according to a certainorder, and the modified transform coefficients are derived therefromthrough the matrix operation with the transform kernel matrix.

According to an example, when performing the matrix operation of thetransform kernel matrix for the 8×8 region, the 48 transformcoefficients in the top-left 4×4 region, the top-right 4×4 region, andthe bottom-left 4×4 region of the 8×8 region may be one-dimensionallyarranged according to either the row-first order or the column-firstorder according to the intra prediction mode applied to the target blockas shown in Table 15 or Table 18, and the derived 16 modified transformcoefficients may be arranged in a diagonal scanning direction in thetop-left 4×4 region of the 8×8 region as shown in Table 16.

As described above, the transformer 232 may apply the transform kernelmatrix to 16 transform coefficients in the 4×4 target block, therebyderiving eight modified transform coefficients corresponding to atop-left region of the 4×4 region. That is, the 16 transformcoefficients in the 4×4 region to be transformed may beone-dimensionally arranged in either a row-first direction or acolumn-first direction according to the intra prediction mode applied tothe target block as shown in Table 12 or Table 13, and the derived eightmodified transform coefficients may be arranged in the diagonal scanningdirection in the top-left region of the 4×4 region.

When an intra prediction mode applicable to the target block is any oneof 65 directional modes, intra prediction modes are symmetric withrespect to intra prediction mode 34 in a top-left diagonal direction,and the intra prediction mode applied to the target block includes modes2 to 34 in a left direction based on intra prediction mode 34, thetransform coefficients in the top-left 4×4 region, the top-right 4×4region, and the bottom-left 4×4 region of the 8×8 region may beone-dimensionally arranged according to the row-first order as shown inTable 15.

When the intra prediction mode applied to the target block includesmodes 35 to 66 in a right direction based on intra prediction mode 34,the transform coefficients in the top-left 4×4 region, the top-right 4×4region, and the bottom-left 4×4 region of the 8×8 region may beone-dimensionally arranged according to the column-first order as shownin Table 17.

When the intra prediction mode applied to the target block is a planarmode or a DC mode, the transform coefficients in the top-left 4×4region, the top-right 4×4 region, and the bottom-left 4×4 region of the8×8 region may be one-dimensionally arranged according to the row-firstorder.

The encoding apparatus may clip the modified transform coefficients tovalues within a predetermined range (S1340).

According to an example, the modified transform coefficients may beclipped based on Equation 9, and the maximum value (maxFwdSecTr) and theminimum value (minFwdSecTr) indicating the clipping range may be set asshown in Equation 11.

In this case, the modified transform coefficients may be scaled andclipped according to Equation 10, and according to an example, S may be7, and in this case, the modified transform coefficients may beconsidered to be scaled by 128.

According to an example, the clipping range applied to the transformcoefficient and the clipping range applied to the modified transformcoefficient may be set to the same range. Alternatively, according toanother example, the clipping range may be set differently according tothe primary transformation and the secondary transformation.

When the non-separated secondary transform is performed in this way, theentropy encoder 240 derives quantized transform coefficients byperforming quantization based on the clipped modified transformcoefficients, and information on the quantized transform coefficientsmay be encoded (S1350).

First, the entropy encoder 240 may derive a syntax element value for thetransform index indicating any one of the transform kernel matricesincluded in the transform set, may binarize the derived syntax elementvalue for the transform index, and may encode bins of a syntax elementbin string based on context information, that is, a context model, on abin string of the transform index.

The encoded bin string of the syntax element may be output as abitstream to the decoding apparatus 300 or to the outside.

In the above-described embodiments, the methods are explained on thebasis of flowcharts by means of a series of steps or blocks, but thepresent disclosure is not limited to the order of steps, and a certainstep may be performed in order or step different from that describedabove, or concurrently with another step. Further, it may be understoodby a person having ordinary skill in the art that the steps shown in aflowchart are not exclusive, and that another step may be incorporatedor one or more steps of the flowchart may be removed without affectingthe scope of the present disclosure.

The above-described methods according to the present disclosure may beimplemented as a software form, and an encoding apparatus and/ordecoding apparatus according to the disclosure may be included in adevice for image processing, such as, a TV, a computer, a smartphone, aset-top box, a display device or the like.

When embodiments in the present disclosure are embodied by software, theabove-described methods may be embodied as modules (processes, functionsor the like) to perform the above-described functions. The modules maybe stored in a memory and may be executed by a processor. The memory maybe inside or outside the processor and may be connected to the processorin various well-known manners. The processor may include anapplication-specific integrated circuit (ASIC), other chipset, logiccircuit, and/or a data processing device. The memory may include aread-only memory (ROM), a random access memory (RAM), a flash memory, amemory card, a storage medium, and/or other storage device. That is,embodiments described in the present disclosure may be embodied andperformed on a processor, a microprocessor, a controller or a chip. Forexample, function units shown in each drawing may be embodied andperformed on a computer, a processor, a microprocessor, a controller ora chip.

Further, the decoding apparatus and the encoding apparatus to which thepresent disclosure is applied, may be included in a multimediabroadcasting transceiver, a mobile communication terminal, a home cinemavideo device, a digital cinema video device, a surveillance camera, avideo chat device, a real time communication device such as videocommunication, a mobile streaming device, a storage medium, a camcorder,a video-on-demand (VoD) service providing device, an over the top (OTT)video device, an Internet streaming service providing device, athree-dimensional (3D) video device, a video telephony video device, anda medical video device, and may be used to process a video signal or adata signal. For example, the over-the-top (OTT) video device mayinclude a game console, a Blu-ray player, an Internet access TV, a Hometheater system, a smartphone, a Tablet PC, a digital video recorder(DVR) and the like.

In addition, the processing method to which the present disclosure isapplied, may be produced in the form of a program executed by acomputer, and be stored in a computer-readable recording medium.Multimedia data having a data structure according to the presentdisclosure may also be stored in a computer-readable recording medium.The computer-readable recording medium includes all kinds of storagedevices and distributed storage devices in which computer-readable dataare stored. The computer-readable recording medium may include, forexample, a Blu-ray Disc (BD), a universal serial bus (USB), a ROM, aPROM, an EPROM, an EEPROM, a RAM, a CD-ROM, a magnetic tape, a floppydisk, and an optical data storage device. Further, the computer-readablerecording medium includes media embodied in the form of a carrier wave(for example, transmission over the Internet). In addition, a bitstreamgenerated by the encoding method may be stored in a computer-readablerecording medium or transmitted through a wired or wirelesscommunication network. Additionally, the embodiments of the presentdisclosure may be embodied as a computer program product by programcodes, and the program codes may be executed on a computer by theembodiments of the present disclosure. The program codes may be storedon a computer-readable carrier.

FIG. 14 illustrates the structure of a content streaming system to whichthe present disclosure is applied.

Further, the contents streaming system to which the present disclosureis applied may largely include an encoding server, a streaming server, aweb server, a media storage, a user equipment, and a multimedia inputdevice.

The encoding server functions to compress to digital data the contentsinput from the multimedia input devices, such as the smart phone, thecamera, the camcorder and the like, to generate a bitstream, and totransmit it to the streaming server. As another example, in a case wherethe multimedia input device, such as, the smart phone, the camera, thecamcorder or the like, directly generates a bitstream, the encodingserver may be omitted. The bitstream may be generated by an encodingmethod or a bitstream generation method to which the present disclosureis applied. And the streaming server may store the bitstream temporarilyduring a process to transmit or receive the bitstream.

The streaming server transmits multimedia data to the user equipment onthe basis of a user's request through the web server, which functions asan instrument that informs a user of what service there is. When theuser requests a service which the user wants, the web server transfersthe request to the streaming server, and the streaming server transmitsmultimedia data to the user. In this regard, the contents streamingsystem may include a separate control server, and in this case, thecontrol server functions to control commands/responses betweenrespective equipments in the content streaming system.

The streaming server may receive contents from the media storage and/orthe encoding server. For example, in a case the contents are receivedfrom the encoding server, the contents may be received in real time. Inthis case, the streaming server may store the bitstream for apredetermined period of time to provide the streaming service smoothly.

For example, the user equipment may include a mobile phone, a smartphone, a laptop computer, a digital broadcasting terminal, a personaldigital assistant (PDA), a portable multimedia player (PMP), anavigation, a slate PC, a tablet PC, an ultrabook, a wearable device(e.g., a watch-type terminal (smart watch), a glass-type terminal (smartglass), a head mounted display (HMD)), a digital TV, a desktop computer,a digital signage or the like. Each of servers in the contents streamingsystem may be operated as a distributed server, and in this case, datareceived by each server may be processed in a distributed manner.

What is claimed is:
 1. An image decoding method performed by a decodingapparatus, the method comprising: receiving quantized transformcoefficients for a target block and a transform index for anon-separable secondary transform; deriving transform coefficients bydequantizing the quantized transform coefficients; deriving the modifiedtransform coefficients based on a matrix operation of a transform kernelmatrix in a transform set related to the transform index and a transformcoefficient list corresponding to a size of the transform coefficients;clipping the modified transform coefficients to values within a specificrange, deriving residual samples for the target block based on aninverse primary transform for clipped modified transform coefficients,and clipping the residual samples to values within a specific range,wherein the deriving the modified transform coefficients derive themodified transform coefficients of the top-left 4×4 region, thetop-right 4×4 region and the bottom-left 4×4 region of the 8×8 region byapplying the transform kernel matrix to the transform coefficients ofthe top-left 4×4 region of the 8×8 region of the target block.
 2. Theimage decoding method of claim 1, wherein the modified transformcoefficients are scaled and clipped according to the following Equation1,y=(x+(1<<(S−1)))>>S   [Equation 1] where S is equal to 7 in Equation 1.3. The image decoding method of claim 1, wherein the maximum value(maxInvSecTr) and the minimum value (minInvSecTr) of the clipping rangeare set by the following Equation 2.maxInvSecTr=2¹⁵−1minInvSecTr=−2¹⁵   [Equation 2]
 4. The image decoding method of claim 1,wherein the transform kernel matrix is a 48×16 matrix, and wherein amatrix operation between the transform kernel matrix and the transformcoefficients of the top-left 4×4 region is (48×16 matrix) * (16×1transform coefficient vector).
 5. The image decoding method of claim 1,further comprising: deriving an input transform coefficient size relatedto a length of the dequantized transform coefficients to which thenon-separable secondary transform is applied and an output transformcoefficient size related to a length of modified transform coefficientsto which the non-separable secondary transform has been applied;deriving the transform set based on a mapping relationship according toan intra prediction mode applied to the target block.
 6. The imagedecoding method of claim 5, further comprising: deriving the transformkernel matrix based on the output transform coefficient sizecorresponding to the number of the modified transform coefficients, thederived transform set, and the transform index .
 7. An image encodingmethod performed by an image encoding apparatus, the method comprising:deriving transform coefficients by dequantizing quantized transformcoefficients; deriving the modified transform coefficients based on amatrix operation of a transform kernel matrix in a transform set and atransform coefficient list corresponding to a size of the transformcoefficients; clipping the modified transform coefficients to valueswithin a specific range, deriving residual samples for the target blockbased on an inverse primary transform for clipped modified transformcoefficients, and clipping the residual samples to values within aspecific range, wherein the deriving the modified transform coefficientsderive the modified transform coefficients of the top-left 4×4 region,the top-right 4×4 region and the bottom-left 4×4 region of the 8×8region by applying the transform kernel matrix to the transformcoefficients of the top-left 4×4 region of the 8×8 region of the targetblock.
 8. The image encoding method of claim 7, wherein the modifiedtransform coefficients are scaled and clipped according to the followingEquation 3,y=(x+(1<<(S−1)))>>S   [Equation 3] where S is equal to 7 in Equation 3.9. The image encoding method of claim 7, wherein the maximum value(maxInvSecTr) and the minimum value (minInvSecTr) of the clipping rangeare set by the following Equation 2.maxInvSecTr=2¹⁵−1minInvSecTr=−2¹⁵   [Equation 2]
 10. The image encoding method of claim7, wherein the transform kernel matrix is a 48×16 matrix, and wherein amatrix operation between the transform kernel matrix and the transformcoefficients of the top-left 4×4 region is (48×16 matrix)*(16×1transform coefficient vector).
 11. The image encoding method of claim 7,further comprising: generating flag information related to whether ornot a transform index is present and the transform index, wherein thetransform index is related to whether the the non-separable secondarytransform is applied to the target block and which one of the transformkernel matrices applied to the target block.
 12. A non-transitorycomputer readable storage medium storing a bitstream generated by amethod, the method comprising: deriving transform coefficients bydequantizing quantized transform coefficients; deriving the modifiedtransform coefficients based on a matrix operation of a transform kernelmatrix in a transform set and a transform coefficient list correspondingto a size of the transform coefficients; clipping the modified transformcoefficients to values within a specific range, deriving residualsamples for the target block based on an inverse primary transform forclipped modified transform coefficients, clipping the residual samplesto values within a specific range, and generating image informationincluding residual information for the residual samples, flaginformation related to whether or not a transform index is present andthe transform index to generate the bitstream, wherein the transformindex is related to whether the the non-separable secondary transform isapplied to the target block and which one of the transform kernelmatrices applied to the target block wherein the deriving the modifiedtransform coefficients derive the modified transform coefficients of thetop-left 4×4 region, the top-right 4×4 region and the bottom-left 4×4region of the 8×8 region by applying the transform kernel matrix to thetransform coefficients of the top-left 4×4 region of the 8×8 region ofthe target block.